Homotopy lifting property: Difference between revisions

Revision as of 23:01, 7 December 2013

In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A. It is usually denoted A^∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the monoid operation is string concatenation. The free semigroup on A is the subsemigroup of A^∗ containing all elements except the empty string. It is usually denoted A⁺.^[1]^[2]

More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set.^[3]

As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.

Examples

Natural numbers

The monoid (N,+) of natural numbers (including zero) under addition is a free monoid on a singleton free generator, in this case the natural number 1. According to the formal definition, this monoid consists of all sequences like "1", "1+1", "1+1+1", "1+1+1+1", and so on, including the empty sequence. Mapping each such sequence to its evaluation result ^[4] and the empty sequence to zero establishes an isomorphism from the set of such sequences to N. This isomorphism is compatible with "+", that is, for any two sequences s and t, if s is mapped (i.e. evaluated) to a number m and t to n, then their concatenation s+t is mapped to the sum m+n.

Kleene star

In formal language theory, usually a finite set (called "alphabet" there) A of "symbols" is considered, a finite sequence of symbols is called "word over A", and the free monoid A^∗ is called the "Kleene star of A". Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids. There are deep connections between the theory of semigroups and that of automata. For example, the regular languages over A are the homomorphic pre-images in A^∗ of subsets of finite monoids.Template:Clarify

For example, assuming an alphabet A = {a, b, c}, its Kleene star A^∗ contains all concatenations of a, b, and c:

{ε, a, ab, ba, caa, cccbabbc, ...}.

If A is any set, the word length function on A^∗ is the unique monoid homomorphism from A^∗ to (N,+) that maps each element of A to 1. A free monoid is thus a graded monoid.Template:Clarify^[5]

Conjugate words

We define a pair of words in A^∗ of the form uv and vu as conjugate: the conjugates of a word are thus its circular shifts.^[6] Two words are conjugate in this sense if they are conjugate in the sense of group theory as elements of the free group generated by A.^[7]

Equidivisibility

A free monoid is equidivisible: if the equation mn = pq holds, then there exists an s such that either m = ps, sn = q (example see image) or ms = p, n = sq.^[8]

A monoid is free if and only if it is graded and equidivisible.^[8]

Free generators and rank

The members of a set A are called the free generators for A^∗ and A⁺. The superscript * is then commonly understood to be the Kleene star. More generally, if S is an abstract free monoid (semigroup), then a set of elements which maps onto the set of single-letter words under an isomorphism to a semigroup A⁺ (monoid A^∗) is called a set of free generators for S.

Each free semigroup (or monoid) S has exactly one set of free generators, the cardinality of which is called the rank of S.

Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, every set of generators for a free semigroup or monoid S contains the free generators.Template:Clarify It follows that a free semigroup or monoid is finitely generated if and only if it has finite rank.

A submonoid N of A^∗ is stable if u, v, ux, xv in N together imply x in N.^[9] A submonoid of A^∗ is stable if and only if it is free.^[10] For example, using the set of bits { "0", "1" } as A, the set N of all bit strings containing evenly many "1"s is a stable^[11] submonoid^[12] of the set A^∗ of all bit strings at all. While N cannot be freely generated by any set of single bits, it can be freely generated by the set of bit strings { "0", "11", "101", "1001", "10001", ... }.Template:Clarify

Codes

A set of free generators for a free monoid P is referred to as a basis for P: a set of words C is a code if C* is a free monoid and C is a basis.^[3] A set X of words in A^∗ is a prefix, or has the prefix property, if it does not contain a proper (string) prefix of any of its elements. Every prefix in A⁺ is a code, indeed a prefix code.^[3]^[13]

A submonoid N of A^∗ is right unitary if x, xy in N implies y in N. A submonoid is generated by a prefix if and only if it is right unitary.^[14]

Free hull

The intersection of free submonoids of a free monoid A^∗ is again free.^[15]^[16] If S is a subset of a free monoid A* then the intersection of all free submonoids of A* containing S is well-defined, since A* itself is free, and contains S; it is a free monoid. A basis for this intersection is the free hull of S.

The defect theorem^[15]^[16]^[17] states that if X is finite and C is the free hull of X, then either X is a code and C = X, or

|C| ≤ |X| − 1 .

Morphisms

A monoid morphism f from a free monoid B^∗ to a monoid M is a map such that f(xy) = f(x)⋅f(y) for words x,y and f(ε) = ι, where ε and ι denotes the identity element of B^∗ and M, respectively. The morphism f is determined by its values on the letters of B and conversely any map from B to M extends to a morphism. A morphism is non-erasing^[18] or continuous^[19] if no letter of B maps to ι and trivial if every letter of B maps to ι.^[20]

A morphism f from a free monoid B^∗ to a free monoid A^∗ is total if every letter of A occurs in some word in the image of f; cyclic^[20] or periodic^[21] if the image of f is contained inTemplate:Clarify w^∗ for some word w of A^∗. A morphism f is k-uniform if the length |f(a)| is constant and equal to k for all a in A.^[22]^[23] A 1-uniform morphism is strictly alphabetic^[19] or a coding.^[24]

A morphism f from a free monoid B^∗ to a free monoid A^∗ is simplifiable if there is an alphabet C of cardinality less than that of B such the morphism f factors through C^∗;Template:Clarify otherwise f is elementary. The morphism f is called a code if the image of the alphabet B under f is a code: every elementary morphism is a code.^[25]

Test sets

For L a subset of B^∗, a finite subset T of L is a test set for L if morphisms f and g on B^∗ agree on L if and only if they agree on T. The Ehrenfeucht conjecture is that any subset L has a test set:^[26] it has been proved^[27] independently by Albert and Lawrence; McNaughton; and Guba. The proofs rely on Hilbert's basis theorem.^[28]

Endomorphisms

An endomorphism of A^∗ is a morphism from A^∗ to itself.^[29] The identity map I is an endomorphism of A^∗, and the endomorphisms form a monoid under composition of functions.

An endomorphism f is prolongable if there is a letter a such that f(a) = as for a non-empty string s.^[30]

String projection

The operation of string projection is an endomorphism. That is, given a letter a ∈ Σ and a string s ∈ Σ^∗, the string projection p_a(s) removes every occurrence of a from s; it is formally defined by

p_{a}(s)={\begin{cases}\varepsilon &{\text{if }}s=\varepsilon ,{\text{ the empty string}}\\p_{a}(t)&{\text{if }}s=ta\\p_{a}(t)b&{\text{if }}s=tb{\text{ and }}b\neq a.\end{cases}}

Note that string projection is well-defined even if the rank of the monoid is infinite, as the above recursive definition works for all strings of finite length. String projection is a morphism in the category of free monoids, so that

p_{a}\left(\Sigma ^{*}\right)=\left(\Sigma -a\right)^{*}

where $p_{a}\left(\Sigma ^{*}\right)$ is understood to be the free monoid of all finite strings that don't contain the letter a. The identity morphism is $p_{\varepsilon }$ ,Template:Clarify as clearly $p_{\varepsilon }(s)=s$ for all strings s. Of course, it commutes with the operation of string concatenation, so that $p_{a}(st)=p_{a}(s)p_{a}(t)$ for all strings s and t. There are many right inverses to string projection, and thus it is a split epimorphism.

String projection is commutative, as clearly

p_{a}(p_{b}(s))=p_{b}(p_{a}(s)).

For free monoids of finite rank, this follows from the fact that free monoids of the same rank are isomorphic, as projection reduces the rank of the monoid by one.

String projection is idempotent, as

p_{a}(p_{a}(s))=p_{a}(s)

for all strings s. Thus, projection is an idempotent, commutative operation, and so it forms a bounded semilattice or a commutative band.

Sturmian endomorphisms

An endomorphism of the free monoid B^∗ on a 2-letter alphabet B is Sturmian if it maps every Sturmian word to a Sturmian word^[31]^[32] and locally Sturmian if it maps some Sturmian word to a Sturmian word.^[33] The Sturmian endomorphisms form a submonoid of the monoid of endomorphisms of B^∗.^[31]

Define endomorphisms φ and ψ of B^∗, where B = {0,1}, by φ(0) = 01, φ(1) = 0 and ψ(0) = 10, ψ(1) = 0. Then I, φ and ψ are Sturmian,^[34] and the Sturmian endomorphisms of B^∗ are precisely those endomorphisms in the submonoid of the endomorphism monoid generated by {I,φ,ψ}.^[35]^[33]^[32]

A primitive substitution is Sturmian if the image of the word 10010010100101 is balanced.Template:Clarify^[36]^[32]

The free commutative monoid

Given a set A, the free commutative monoid on A is the set of all finite multisets with elements drawn from A, with the monoid operation being multiset sum and the monoid unit being the empty multiset.

For example, if A = {a, b, c}, elements of the free commutative monoid on A are of the form

{ε, a, ab, a²b, ab³c⁴, ...}.

The fundamental theorem of arithmetic states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers.

The free commutative semigroup is the subset of the free commutative monoid which contains all multisets with elements drawn from A except the empty multiset.

Generalization

The free partially commutative monoid, or trace monoid, is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in combinatorics and in the study of parallelism in computer science.

Free monoids and computing

The free monoid on a set A corresponds to lists of elements from A with concatenation as the binary operation. A monoid homomorphism from the free monoid to any other monoid (M,•) is a function f such that

f(x₁…x_n) = f(x₁) • … • f(x_n)
f() = e

where e is the identity on M. Computationally, every such homomorphism corresponds to a map operation applying f to all the elements of a list, followed by a fold operation which combines the results using the binary operator •. This computational paradigm (which can be generalised to non-associative binary operators) has inspired the MapReduce software framework.

Notes

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References

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

↑ Template:Harvtxt, [1]
↑ Template:Harvtxt
↑ ^3.0 ^3.1 ^3.2 Template:Harvtxt
↑ Since addition of natural numbers is associative, the result doesn't depend on the order of evaluation, thus ensuring the mapping to be well-defined.
↑ Sakarovitch (2009) p.382
↑ Sakarovitch (2009) p.27
↑ Template:Harvtxt
↑ ^8.0 ^8.1 Sakarovitch (2009) p.26
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ if u contains an even number of "1"s, and ux as well, then x must contain an even number of "1"s, too
↑ since it is closed with respect to string concatenation
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ ^15.0 ^15.1 Template:Harvtxt
↑ ^16.0 ^16.1 Template:Harvtxt
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ ^19.0 ^19.1 Template:Harvtxt
↑ ^20.0 ^20.1 Template:Harvtxt
↑ Salomaa (1981) p.77
↑ Template:Harvtxt
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ Template:Harvtxt
↑ Salomaa (1981) p.72
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Template:Harvtxt
↑ Allouche & Shallit (2003) p.10
↑ ^31.0 ^31.1 Template:Harvtxt
↑ ^32.0 ^32.1 ^32.2 Template:Harvtxt
↑ ^33.0 ^33.1 Template:Harvtxt
↑ Template:Harvtxt
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[Lot23-1] Template:Harvtxt, [1]

[PF2-2] Template:Harvtxt

[Lot5-3] 3.0 ^3.1 ^3.2 Template:Harvtxt

[4] Since addition of natural numbers is associative, the result doesn't depend on the order of evaluation, thus ensuring the mapping to be well-defined.

[Sak382-5] Sakarovitch (2009) p.382

[Sak27-6] Sakarovitch (2009) p.27

[PF297-7] Template:Harvtxt

[Sak26-8] 8.0 ^8.1 Sakarovitch (2009) p.26

[BPR61-9] Template:Harvtxt

[BPR62-10] Template:Harvtxt

[11] u contains an even number of "1"s, and ux as well, then x must contain an even number of "1"s, too

[12] since it is closed with respect to string concatenation

[BPR58-13] Template:Harvtxt

[Lot15-14] Template:Harvtxt

[Lot6-15] 15.0 ^15.1 Template:Harvtxt

[LotII204-16] 16.0 ^16.1 Template:Harvtxt

[BPR66-17] Template:Harvtxt

[Lot7-18] Template:Harvtxt

[Sak25-19] 19.0 ^19.1 Template:Harvtxt

[Lot164-20] 20.0 ^20.1 Template:Harvtxt

[Sal77-21] Salomaa (1981) p.77

[ApCoW522-22] Template:Harvtxt

[BR103-23] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[AS9-24] Template:Harvtxt

[Sal72-25] Salomaa (1981) p.72

[Lot1789-26] Template:Harvtxt

[LotII451-27] Template:Harvtxt

[28] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[LotII450-29] Template:Harvtxt

[AS10-30] Allouche & Shallit (2003) p.10

[LotII83-31] 31.0 ^31.1 Template:Harvtxt

[PF197-32] 32.0 ^32.1 ^32.2 Template:Harvtxt

[LotII85-33] 33.0 ^33.1 Template:Harvtxt

[LotII84-34] Template:Harvtxt

[BS1994-35] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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+In [[abstract algebra]], the '''free monoid''' on a [[Set (mathematics)|set]] ''A'' is the [[monoid]] whose elements are all the finite sequences (or [[character string|strings]]) of zero or more elements from ''A''. It is usually denoted ''A''<sup>&lowast;</sup>. The [[identity element]] is the unique sequence of zero elements, often called the [[empty string]] and denoted by ε or λ, and the monoid operation is [[string concatenation]]. The '''free [[semigroup]]''' on ''A'' is the subsemigroup of ''A''<sup>&lowast;</sup> containing all elements except the empty string. It is usually denoted ''A''<sup>+</sup>.<ref name=Lot23>{{harvtxt|Lothaire|1997|pp=2–3}}, [http://books.google.com/books?id=eATLTZzwW-sC&pg=PA2]</ref><ref name=PF2>{{harvtxt|Pytheas Fogg|2002|p=2}}</ref>
+More generally, an abstract monoid (or semigroup) ''S'' is described as '''free''' if it is [[isomorphic]] to the free monoid (or semigroup) on some set.<ref name=Lot5>{{harvtxt|Lothaire|1997|p=5}}</ref>
+As the name implies, free monoids and semigroups are those objects which satisfy the usual [[universal property]] defining [[free object]]s, in the respective [[category (mathematics)|categories]] of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.
+== Examples ==
+=== Natural numbers ===
+The monoid ('''N''',+) of [[natural numbers]] (including zero) under addition is a free monoid on a singleton free generator, in this case the natural number 1.
+According to the formal definition, this monoid consists of all sequences like "1", "1+1", "1+1+1", "1+1+1+1", and so on, including the empty sequence.
+Mapping each such sequence to its evaluation result
+<ref>Since addition of natural numbers is associative, the result doesn't depend on the order of evaluation, thus ensuring the mapping to be well-defined.</ref>
+and the empty sequence to zero establishes an isomorphism from the set of such sequences to '''N'''.
+This isomorphism is compatible with "+", that is, for any two sequences ''s'' and ''t'', if ''s'' is mapped (i.e. evaluated) to a number ''m'' and ''t'' to ''n'', then their concatenation ''s''+''t'' is mapped to the sum ''m''+''n''.
+=== Kleene star ===
+In [[formal language]] theory, usually a finite set (called "alphabet" there) ''A'' of "symbols" is considered, a finite sequence of symbols is called "word over ''A''", and the free monoid ''A''<sup>&lowast;</sup> is called the "[[Kleene star]] of ''A''".
+Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids. There are deep connections between the theory of semigroups and that of [[automata theory|automata]]. For example, the [[regular language]]s over ''A'' are the homomorphic pre-images in ''A''<sup>&lowast;</sup> of subsets of finite monoids.{{clarify|reason=At least, a reference to a definition of 'homomorphic pre-image' and a reference to a proof should be added here. Also, rephrasing the sentence in singular may improve clarity.|date=August 2013}}
+For example, assuming an alphabet ''A'' = {''a'', ''b'', ''c''}, its Kleene star ''A''<sup>&lowast;</sup> contains all concatenations of ''a'', ''b'', and ''c'':
+:{ε, ''a'', ''ab'', ''ba'', ''caa'', ''cccbabbc'', ...}.
+If ''A'' is any set, the ''word length'' function on ''A''<sup>&lowast;</sup> is the unique [[monoid homomorphism]] from ''A''<sup>&lowast;</sup> to ('''N''',+) that maps each element of ''A'' to 1.  A free monoid is thus a '''graded monoid'''.{{clarify|reason=Give a definition of 'graded monoid' or a reference to it. A Wikipedia article 'graded ring' exists, but it doesn't define graded monoids. In case 'graded' is meant just informally, the sentence should better be rephrased as e.g. 'Due to this additional functionality, some authors call a free monoid a ⟨graded⟩ one.'|date=August 2013}}<ref name=Sak382>Sakarovitch (2009) p.382</ref>
+==Conjugate words==
+[[File:Example of strings equidivisibility.gif|thumb|Example for 1st case of equidivisibility: m="UNCLE", n="ANLY", p="UN", q="CLEANLY", and s="CLE"]]
+We define a pair of words in ''A''<sup>&lowast;</sup> of the form ''uv'' and ''vu'' as '''conjugate''': the conjugates of a word are thus its [[circular shift]]s.<ref name=Sak27>Sakarovitch (2009) p.27</ref>  Two words are conjugate in this sense if they are [[Conjugation (group theory)|conjugate in the sense of group theory]] as elements of the [[free group]] generated by ''A''.<ref name=PF297>{{harvtxt|Pytheas Fogg|2002|p=297}}</ref>
+===Equidivisibility===
+A free monoid is '''equidivisible''': if the equation ''mn'' = ''pq'' holds, then there exists an ''s'' such that either ''m'' = ''ps'', ''sn'' = ''q'' (example see image) or ''ms'' = ''p'', ''n'' = ''sq''.<ref name=Sak26>Sakarovitch (2009) p.26</ref>
+A monoid is free if and only if it is graded and equidivisible.<ref name=Sak26/>
+== Free generators and rank ==
+The members of a set ''A'' are called the '''free generators''' for ''A''<sup>&lowast;</sup> and ''A''<sup>+</sup>. The superscript * is then commonly understood to be the [[Kleene star]]. More generally, if ''S'' is an abstract free monoid (semigroup), then a set of elements which maps onto the set of single-letter words under an isomorphism to a semigroup ''A<sup>+</sup>'' (monoid ''A''<sup>&lowast;</sup>) is called a ''set of free generators'' for ''S''.
+Each free semigroup (or monoid) ''S'' has exactly one set of free generators, the [[cardinality]] of which is called the ''rank'' of ''S''.
+Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, ''every'' set of generators for a free semigroup or monoid ''S'' contains the free generators.{{clarify|reason=What is the difference between a set of generators and a set of free generators? Up to here, only the latter has been defined.|date=August 2013}} It follows that a free semigroup or monoid is finitely generated if and only if it has finite rank.
+A [[Submonoid#Generators_and_submonoids|submonoid]] ''N'' of ''A''<sup>&lowast;</sup> is '''stable''' if ''u'', ''v'', ''ux'', ''xv'' in ''N'' together imply ''x'' in ''N''.<ref name=BPR61>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=61}}</ref>  A submonoid of ''A''<sup>&lowast;</sup> is stable if and only if it is free.<ref name=BPR62>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=62}}</ref>
+For example, using the set of [[bit]]s { "0", "1" } as ''A'', the set ''N'' of all bit strings containing evenly many "1"s is a stable<ref>if ''u'' contains an even number of "1"s, and ''ux'' as well, then ''x'' must contain an even number of "1"s, too</ref> submonoid<ref>since it is closed with respect to string concatenation</ref> of the set ''A''<sup>&lowast;</sup> of all bit strings at all. While ''N'' cannot be freely generated by any set of single bits, it ''can'' be freely generated by the set of bit strings { "0", "11", "101", "1001", "10001", ... }.{{clarify|reason=Somebody please cross-check my example: Aren't generator sets restricted to single-character sets, so my generator set wouldn't be valid? But I can't imagine another one. On the other hand, each member of N can obviously be decomposed into a sequence of my generators in a unique way, e.g. 1010101 → 101⋅0⋅101|date=August 2013}}
+===Codes===
+A set of free generators for a free monoid ''P'' is referred to as a '''basis''' for '''P''': a set of words ''C'' is a '''code''' if ''C''* is a free monoid and ''C'' is a basis.<ref name=Lot5/>  A set ''X'' of words in ''A''<sup>&lowast;</sup> is a '''prefix''', or has the '''prefix property''', if it does not contain a proper [[prefix (computer science)|(string) prefix]] of any of its elements<!--- deleted, since "≤" as shorthand for "string prefix of" has not been defined yet, and is not used elsewhere in the article: that is, ''x'',''y'' in ''X'' with ''x'' ≤ ''y'' implies ''x'' = ''y''--->.  Every prefix in ''A''<sup>+</sup> is a code, indeed a [[prefix code]].<ref name=Lot5/><ref name=BPR58>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=58}}</ref>
+A submonoid ''N'' of ''A''<sup>&lowast;</sup> is '''right unitary''' if ''x'', ''xy'' in ''N'' implies ''y'' in ''N''.  A submonoid is generated by a prefix if and only if it is right unitary.<ref name=Lot15>{{harvtxt|Lothaire|1997|p=15}}</ref>
+==Free hull==
+The intersection of free submonoids of a free monoid ''A''<sup>&lowast;</sup> is again free.<ref name=Lot6>{{harvtxt|Lothaire|1997|p=6}}</ref><ref name=LotII204>{{harvtxt|Lothaire|2011|p=204}}</ref>  If ''S'' is a subset of a free monoid ''A''* then the intersection of all free submonoids of ''A''* containing ''S'' is well-defined, since ''A''* itself is free, and contains ''S''; it is a free monoid.  A basis for this intersection is the '''free hull''' of ''S''.
+The '''defect theorem'''<ref name=Lot6/><ref name=LotII204/><ref name=BPR66>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=66}}</ref> states that if ''X'' is finite and ''C'' is the free hull of ''X'', then either ''X'' is a code and ''C'' = ''X'', or
+:|''C''| ≤ |''X''| − 1 .
+==Morphisms==
+A [[monoid morphism]] ''f'' from a free monoid ''B''<sup>&lowast;</sup> to a monoid ''M'' is a map such that ''f''(''xy'') = ''f''(''x'')⋅''f''(''y'') for words ''x'',''y'' and ''f''(ε) = ι, where ε and ι denotes the identity element of ''B''<sup>&lowast;</sup> and ''M'', respectively.  The morphism ''f'' is determined by its values on the letters of ''B'' and conversely any map from ''B'' to ''M'' extends to a morphism.  A morphism is '''non-erasing'''<ref name=Lot7>{{harvtxt|Lothaire|1997|p=7}}</ref> or '''continuous'''<ref name=Sak25>{{harvtxt|Sakarovitch|2009|p=25}}</ref> if no letter of ''B'' maps to ι and '''trivial''' if every letter of ''B'' maps to ι.<ref name=Lot164>{{harvtxt|Lothaire|1997|p=164}}</ref>
+A morphism ''f'' from a free monoid ''B''<sup>&lowast;</sup> to a free monoid ''A''<sup>&lowast;</sup> is '''total''' if every letter of ''A'' occurs in some word in the image of ''f''; '''cyclic'''<ref name=Lot164/> or '''periodic'''<ref name=Sal77>Salomaa (1981) p.77</ref> if the image of ''f'' is contained in{{clarify|reason=The image of f is a set of strings, while w° is a string. None of the meanings of 'is contained in' fits, i.e. neither 'is an element of', nor 'is a subset of', nor 'is a substring of'. Probably, every value f(v) is required to be a substring of w°.|date=August 2013}} ''w''<sup>&lowast;</sup> for some word ''w'' of ''A''<sup>&lowast;</sup>.  A morphism ''f'' is '''''k''-uniform''' if the length |''f''(''a'')| is constant and equal to ''k'' for all ''a'' in ''A''.<ref name=ApCoW522>{{harvtxt|Lothaire|2005|p=522}}</ref><ref name=BR103>{{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 | page=103 }}</ref>  A 1-uniform morphism is '''strictly alphabetic'''<ref name=Sak25/> or a '''coding'''.<ref name=AS9>{{harvtxt|Allouche|Shallit|2003|p=9}}</ref>
+A morphism ''f'' from a free monoid ''B''<sup>&lowast;</sup> to a free monoid ''A''<sup>&lowast;</sup> is '''simplifiable''' if there is an alphabet ''C'' of cardinality less than that of ''B'' such the morphism ''f'' factors through ''C''<sup>&lowast;</sup>;{{clarify|reason=Please give a definition of 'factors through' or a link to it.|date=August 2013}} otherwise ''f'' is '''elementary'''.  The morphism ''f'' is called a '''code''' if the image of the alphabet ''B'' under ''f'' is a code: every elementary morphism is a code.<ref name=Sal72>Salomaa (1981) p.72</ref>
+===Test sets===
+For  ''L'' a subset of ''B''<sup>&lowast;</sup>, a finite subset ''T'' of ''L'' is a ''test set'' for ''L'' if morphisms ''f'' and ''g'' on ''B''<sup>&lowast;</sup> agree on ''L'' if and only if they agree on ''T''.  The '''Ehrenfeucht conjecture''' is that any subset ''L'' has a test set:<ref name=Lot1789>{{harvtxt|Lothaire|1997|pp=178–179}}</ref> it has been proved<ref name=LotII451>{{harvtxt|Lothaire|2011|p=451}}</ref> independently by Albert and Lawrence; McNaughton; and Guba.  The proofs rely on [[Hilbert's basis theorem]].<ref>{{cite journal | first=A. | last=Salomaa | authorlink=Arto Salomaa | title=The Ehrenfeucht conjecture: A proof for language theorists | journal=Bulletin of the EATCS | year=1985 | number=27 | date=October 1985 | pages=71–82 }}</ref>
+==Endomorphisms==
+An '''[[endomorphism]]''' of ''A''<sup>&lowast;</sup> is a morphism from ''A''<sup>&lowast;</sup> to itself.<ref name=LotII450>{{harvtxt|Lothaire|2011|p=450}}</ref>  The [[identity map]] ''I'' is an endomorphism of ''A''<sup>&lowast;</sup>, and the endomorphisms form a [[monoid]] under [[composition of functions]].
+An endomorphism ''f'' is '''prolongable''' if there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s''.<ref name=AS10>Allouche & Shallit (2003) p.10</ref>
+===String projection===
+The operation of [[String_operations#String_projection|string projection]] is an endomorphism. That is, given a letter ''a'' &isin; &Sigma; and a string ''s'' &isin; &Sigma;<sup>&lowast;</sup>, the string projection ''p''<sub>a</sub>(''s'') removes every occurrence of ''a'' from ''s''; it is formally defined by
+:<math>p_a(s) = \begin{cases}
+\varepsilon & \text{if } s=\varepsilon, \text{ the empty string} \\
+p_a(t) & \text{if } s=ta  \\
+p_a(t)b & \text{if } s=tb \text{ and } b\ne a.
+\end{cases}</math>
+Note that string projection is well-defined even if the rank of the monoid is infinite, as the above recursive definition works for all strings of finite length. String projection is a [[morphism]] in the category of free monoids, so that
+:<math>p_a\left(\Sigma^*\right)= \left(\Sigma-a\right)^*</math>
+where <math>p_a\left(\Sigma^*\right)</math> is understood to be the free monoid of all finite strings that don't contain the letter ''a''.  The identity morphism is <math>p_\varepsilon</math>,{{clarify|reason=Up to here, string projection has been defined only for single letters, not for arbitrary strings. However, the empty string ε is not a letter.|date=August 2013}} as clearly <math>p_\varepsilon(s)=s</math> for all strings ''s''. Of course, it commutes with the operation of string concatenation, so that <math>p_a(st)=p_a(s)p_a(t)</math> for all strings ''s'' and ''t''. There are many right inverses to string projection, and thus it is a [[split epimorphism]].
+String projection is commutative, as clearly
+:<math>p_a(p_b(s))=p_b(p_a(s)).</math>
+For free monoids of finite rank, this follows from the fact that free monoids of the same rank are isomorphic, as projection reduces the rank of the monoid by one.
+String projection is [[idempotent]], as
+:<math>p_a(p_a(s))=p_a(s)</math>
+for all strings ''s''. Thus, projection is an idempotent, commutative operation, and so it forms a bounded [[semilattice]] or a commutative [[band (algebra)|band]].
+===Sturmian endomorphisms===
+An endomorphism of the free monoid ''B''<sup>&lowast;</sup> on a 2-letter alphabet ''B'' is '''Sturmian''' if it maps every [[Sturmian word]] to a Sturmian word<ref name=LotII83>{{harvtxt|Lothaire|2011|p=83}}</ref><ref name=PF197>{{harvtxt|Pytheas Fogg|2002|p=197}}</ref> and '''locally Sturmian''' if it maps some Sturmian word to a Sturmian word.<ref name=LotII85>{{harvtxt|Lothaire|2011|p=85}}</ref>  The Sturmian endomorphisms form a submonoid of the monoid of endomorphisms of ''B''<sup>&lowast;</sup>.<ref name=LotII83/>
+Define endomorphisms φ and ψ of ''B''<sup>&lowast;</sup>, where ''B'' = {0,1}, by φ(0) = 01, φ(1) = 0 and ψ(0) = 10, ψ(1) = 0.  Then ''I'', φ and ψ are Sturmian,<ref name=LotII84>{{harvtxt|Lothaire|2011|p=84}}</ref> and the Sturmian endomorphisms of ''B''<sup>&lowast;</sup> are precisely those endomorphisms in the submonoid of the endomorphism monoid generated by {''I'',φ,ψ}.<ref name=BS1994>{{cite journal | last1=Berstel | first1=J. | last2=Séébold | first2=P. | title=A remark on morphic Sturmian words | journal=RAIRO, Inform. Théor. Appl. 2| volume=8 | number=3-4 | pages=255–263 | year=1994 | issn=0988-3754 | zbl=0883.68104 }}</ref><ref name=LotII85/><ref name=PF197/>
+A primitive substitution is Sturmian if the image of the word 10010010100101 is balanced.{{clarify|reason=Define 'primitive substitution' and 'balanced'; indicate whether the notion of a 'sturmian substitution' is subsumed by the definition of a 'sturmian endomorphism' or whether is something different.|date=August 2013}}<ref name=BS1993>{{citation | chapter=A characterization of Sturmian morphisms | first1=Jean | last1=Berstel | first2=Patrice | last2=Séébold | title=Mathematical Foundations of Computer Science 1993.  18th International Symposium, MFCS'93 Gdańsk, Poland, August 30–September 3, 1993 Proceedings | editor1-first=Andrzej M. | editor1-last=Borzyszkowski | editor2-first=Stefan | editor2-last=Sokołowski | pages=281–290 | year=1993 | doi=10.1007/3-540-57182-5_20 | isbn=978-3-540-57182-7 | series=Lecture Notes in Computer Science | volume=711 | zbl=0925.11026 }}</ref><ref name=PF197/>
+==The free commutative monoid==
+Given a set ''A'', the '''free [[commutative monoid]]''' on ''A'' is the set of all finite [[multiset]]s with elements drawn from ''A'', with the monoid operation being multiset sum and the monoid unit being the empty multiset.
+For example, if ''A'' = {''a'', ''b'', ''c''}, elements of the free commutative monoid on ''A'' are of the form
+:{ε, ''a'', ''ab'', ''a''<sup>2</sup>''b'', ''ab''<sup>3</sup>''c''<sup>4</sup>, ...}.
+The [[fundamental theorem of arithmetic]] states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the [[prime number]]s.
+The '''free commutative semigroup''' is the subset of the free commutative monoid which contains all multisets with elements drawn from ''A'' except the empty multiset.
+==Generalization==
+The [[free partially commutative monoid]], or ''[[trace monoid]]'', is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in [[combinatorics]] and in the study of [[Parallel computing|parallelism]] in [[computer science]].
+==Free monoids and computing==
+The free monoid on a set ''A'' corresponds to [[List (computing)|lists]] of elements from ''A'' with concatenation as the binary operation.  A [[monoid homomorphism]] from the free monoid to any other monoid (''M'',&bull;) is a function ''f'' such that
+* ''f''(''x''<sub>1</sub>…''x''<sub>''n''</sub>) = ''f''(''x''<sub>1</sub>) &bull; … &bull; ''f''(''x''<sub>''n''</sub>)
+* ''f''() = ''e''
+where ''e'' is the identity on ''M''.  Computationally, every such homomorphism corresponds to a [[Map (higher-order function)|map]] operation applying ''f'' to all the elements of a list, followed by a [[Fold (higher-order function)|fold]] operation which combines the results using the binary operator &bull;. This [[squiggol|computational paradigm]] (which can be generalised to non-associative binary operators) has inspired the [[MapReduce]] software framework.
+==See also==
+* [[String operations]]
+==Notes==
+{{reflist}}
+==References==
+* {{citation | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
+* {{citation | last1=Berstel | first1=Jean | last2=Perrin | first2=Dominique | last3=Reutenauer | first3=Christophe | title=Codes and automata | series=Encyclopedia of Mathematics and its Applications | volume=129 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-88831-8 | zbl=1187.94001 }}
+*{{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | others=Contributors: Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R. Series editors: Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon | title=Combinatorics on words | edition=2nd | series=Cambridge Mathematical Library | volume=17 | publisher=[[Cambridge University Press]] | year=1997 | doi = 10.1017/CBO9780511566097 | isbn=0-521-59924-5 | mr = 1475463  | zbl=0874.20040 }}
+* {{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
+*{{citation | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Applied combinatorics on words | others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé| series=Encyclopedia of Mathematics and Its Applications | volume=105 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2005 | isbn=0-521-84802-4 | zbl=1133.68067 }}
+* {{citation | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
+* {{citation | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | location=Cambridge | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
+* {{citation | first=Arto | last=Salomaa | authorlink=Arto Salomaa | title=Jewels of Formal Language Theory | publisher=Pitman Publishing | isbn=0-273-08522-0 | year=1981 | zbl=0487.68064 }}
+[[Category:Semigroup theory]]
+[[Category:Formal languages]]
+[[Category:Free algebraic structures]]
+[[Category:Combinatorics on words]]

Homotopy lifting property: Difference between revisions

Revision as of 23:01, 7 December 2013

Contents

Examples

Natural numbers

Kleene star

Conjugate words

Equidivisibility

Free generators and rank

Codes

Free hull

Morphisms

Test sets

Endomorphisms

String projection

Sturmian endomorphisms

The free commutative monoid

Generalization

Free monoids and computing

See also

Notes

References

Navigation menu

Homotopy lifting property: Difference between revisions

Revision as of 23:01, 7 December 2013

Examples

Natural numbers

Kleene star

Conjugate words

Equidivisibility

Free generators and rank

Codes

Free hull

Morphisms

Test sets

Endomorphisms

String projection

Sturmian endomorphisms

The free commutative monoid

Generalization

Free monoids and computing

See also

Notes

References

Navigation menu

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