|
|
| Line 1: |
Line 1: |
| {{More footnotes|date=May 2009}}
| | Name: Bryant Suggs<br>My age: 40<br>Country: Switzerland<br>Home town: Agasul <br>Postal code: 8308<br>Street: Postfach 50<br><br>my blog - [http://ebookpdfree.wordpress.com/2014/08/26/looking-for-alaska-pdf-free-download/ Looking for Alaska pdf] |
| {{see also|Lattice (group)}}
| |
| {{TOCright}}
| |
| In [[mathematics]], a '''lattice''' is a [[partially ordered set]] in which every two elements have a [[supremum]] (also called a least upper bound or [[Join and meet|join]]) and an [[infimum]] (also called a greatest lower bound or [[Join and meet|meet]]). An example is given by the [[natural number]]s, partially ordered by [[divisibility]], for which the supremum is the [[least common multiple]] and the infimum is the [[greatest common divisor]].
| |
| | |
| Lattices can also be characterized as [[algebraic structure]]s satisfying certain [[axiom]]atic [[Identity (mathematics)|identities]]. Since the two definitions are equivalent, lattice theory draws on both [[order theory]] and [[universal algebra]]. [[Semilattice]]s include lattices, which in turn include [[Heyting algebra|Heyting]] and [[Boolean algebra (structure)|Boolean algebra]]s. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.
| |
| | |
| {{Algebraic structures |Lattice}}
| |
| {{Binary relations}}
| |
| | |
| == Lattices as partially ordered sets ==
| |
| <!---joined into examples section: [[File:Lattice of partitions of an order 4 set.svg|thumb|360px|The name "lattice" is suggested by the form of the [[Hasse diagram]] depicting it. Shown here is the lattice of [[partition (set theory)|partition]]s of a four-element set {1,2,3,4}, ordered by the relation "is a refinement of".]]--->
| |
| | |
| If (''L'', ≤) is a [[partially ordered set]] (poset), and ''S''⊆''L'' is an arbitrary subset, then an element ''u''∈''L'' is said to be an '''upper bound''' of ''S'' if
| |
| ''s''≤''u'' for each ''s''∈''S''. A set may have many upper bounds, or none at all. An upper bound ''u'' of ''S'' is said to be its '''least upper bound''', or '''[[Join (mathematics)|join]]''', or '''supremum''', if ''u''≤''x'' for each upper bound ''x'' of ''S''. A set need not have a least upper bound, but it cannot have more than one. Dually, ''l''∈''L'' is said to be a '''lower bound''' of ''S'' if ''l''≤''s'' for each ''s''∈''S''. A lower bound ''l'' of ''S'' is said to be its '''greatest lower bound''', or '''[[meet (mathematics)|meet]]''', or '''infimum''', if ''x''≤''l'' for each lower bound ''x'' of ''S''. A set may have many lower bounds, or none at all, but can have at most one greatest lower bound.
| |
| | |
| A partially ordered set (''L'', ≤) is called a '''[[join-semilattice]]''' and a '''[[meet-semilattice]]''' if each two-element subset {''a'',''b''} ⊆ ''L'' has a join (i.e. least upper bound) and a meet (i.e. greatest lower bound), denoted by ''a''∨''b'' and ''a''∧''b'', respectively. (''L'', ≤) is called a '''lattice''' if it is both a join- and a meet-semilattice.
| |
| This definition makes ∨ and ∧ [[binary operation]]s. Both operations are monotone with respect to the order: ''a''<sub>1</sub> ≤ ''a''<sub>2</sub> and ''b''<sub>1</sub> ≤ ''b''<sub>2</sub> implies that a<sub>1</sub>∨ b<sub>1</sub> ≤ a<sub>2</sub> ∨ b<sub>2</sub> and a<sub>1</sub>∧b<sub>1</sub> ≤ a<sub>2</sub>∧b<sub>2</sub>.
| |
| | |
| It follows by an [[mathematical induction|induction]] argument that every non-empty finite subset of a lattice has a join and a meet. With additional assumptions, further conclusions may be possible; ''see'' [[Completeness (order theory)]] for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable [[Galois connection]]s between related partially ordered sets — an approach of special interest for the [[category theory|category theoretic]] approach to lattices.
| |
| | |
| A '''bounded lattice''' is a lattice that additionally has a '''[[greatest element|greatest]]''' element 1 and a '''[[least element|least]]''' element 0, which satisfy
| |
| : 0≤''x''≤1 for every ''x'' in ''L''.
| |
| The greatest and least element is also called the '''maximum''' and '''minimum''', or the '''top''' and '''bottom''' element, and denoted by ⊤ and ⊥, respectively. Every lattice can be converted into a bounded lattice by adding an artificial greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by <math>\bigvee L=a_1\lor\cdots\lor a_n</math> (resp.<math>\bigwedge L=a_1\land\cdots\land a_n</math>) where <math>L=\{a_1,\ldots,a_n\}</math>.
| |
| | |
| A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element ''x'' of a poset it is trivially true (it is a [[vacuous truth]]) that
| |
| <math>\forall a\in\varnothing : x \le a</math> and
| |
| <math>\forall a\in\varnothing : a \le x</math>, and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element <math>\bigvee\varnothing=0</math>, and the meet of the empty set is the greatest element <math>\bigwedge\varnothing=1</math>. This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets ''A'' and ''B'' of a poset ''L'', | |
| | |
| :<math>\bigvee \left( A \cup B \right)= \left( \bigvee A \right) \vee \left( \bigvee B \right)</math> | |
| | |
| and
| |
| | |
| :<math>\bigwedge \left( A \cup B \right)= \left(\bigwedge A \right) \wedge \left( \bigwedge B \right)</math> | |
| | |
| hold. Taking ''B'' to be the empty set,
| |
| | |
| :<math>\bigvee \left( A \cup \emptyset \right) | |
| = \left( \bigvee A \right) \vee \left( \bigvee \emptyset \right)
| |
| = \left( \bigvee A \right) \vee 0
| |
| = \bigvee A</math>
| |
| | |
| and
| |
| | |
| :<math>\bigwedge \left( A \cup \emptyset \right) | |
| = \left( \bigwedge A \right) \wedge \left( \bigwedge \emptyset \right)
| |
| = \left( \bigwedge A \right) \wedge 1
| |
| = \bigwedge A</math>
| |
| | |
| which is consistent with the fact that <math>A \cup \emptyset = A</math>.
| |
| | |
| A lattice element ''y'' is said to '''[[covering relation|cover]]''' another element ''x'', if ''y''>''x'', but there does not exist a ''z'' such that ''y''>''z''>''x''.
| |
| Here, ''y''>''x'' means ''x'' ≤ ''y'' and ''x'' ≠ ''y''.
| |
| | |
| A lattice (''L'',≤) is called '''[[Graded poset|graded]]''', sometimes '''ranked''' (but see [[ranked poset|this article]] for an alternative meaning), if it can be equipped with a '''rank function''' ''r'' from ''L'' to ℕ, sometimes to ℤ, compatible with the ordering (so ''r''(''x'')<''r''(''y'') whenever ''x''<''y'') such that whenever ''y'' covers ''x'', then ''r''(''y'')=''r''(''x'')+1. The value of the rank function for a lattice element is called its '''rank'''.
| |
| | |
| Given a subset of a lattice, <math>H \subset L</math>, meet and join restrict to [[partial function]]s – they are undefined if their value is not in the subset <math>H</math>. The resulting structure on <math>H</math> is called a '''{{visible anchor|partial lattice}}'''. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.{{sfn|Grätzer|1996|p=[http://books.google.com/books?id=SoGLVCPuOz0C&pg=PA52 52]}}
| |
| | |
| == Lattices as algebraic structures ==
| |
| | |
| ===General lattice===
| |
| An [[algebraic structure]] (''L'', <math>\lor, \land</math>), consisting of a set ''L'' and two binary [[Operation (mathematics)|operations]] <math>\lor</math>, and <math>\land</math>, on ''L'' is a '''lattice''' if the following axiomatic identities hold for all elements ''a, b, c'' of ''L''.
| |
| | |
| {| style="margin:0em" cellpadding=0 border=0 cellspacing=0
| |
| |
| |
| ;[[commutative property|Commutative laws]]
| |
| :<math>a \lor b = b \lor a</math>,
| |
| :<math>a \land b = b\land a</math>.
| |
| |
| |
| |
| |
| ;[[Associative property|Associative laws]]
| |
| :<math>a \lor(b \lor c) = (a \lor b)\lor c</math>,
| |
| :<math>a \land(b \land c) = (a \land b)\land c</math>.
| |
| |
| |
| |
| |
| ;[[Absorption law]]s:
| |
| :<math>a \lor(a \land b) = a</math>,
| |
| :<math>a \land (a \lor b) = a</math>.
| |
| |}
| |
| | |
| The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.<ref group=note><math>a \lor a=a\lor(a\land(a\lor a))=a</math>, and dually for the other idempotent law. {{Citation | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | title=Ueber Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler | year=1897 | journal=Braunschweiger Festschrift | pages=1–40}}.</ref>
| |
| ;[[Idempotence|Idempotent laws]]
| |
| :<math>a \lor a = a</math>,
| |
| :<math>a \land a = a</math>.
| |
| | |
| These axioms assert that both (''L'',<math>\lor</math>) and (''L'',<math> \land</math>) are [[semilattice]]s. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattices and assure that the two semilattices interact appropriately. In particular, each semilattice is the [[Duality (order theory)|dual]] of the other.
| |
| | |
| ===Bounded lattice===
| |
| A '''bounded lattice''' is an algebraic structure of the form (''L'', <math> \lor, \land</math>, 1, 0) such that (''L'', <math> \lor, \land</math>) is a lattice, 0 (the lattice's bottom) is the [[identity element]] for the join operation <math> \lor</math>, and 1 (the lattice's top) is the identity element for the meet operation <math> \land</math>.
| |
| | |
| ;[[Identity (mathematics)|Identity laws]]
| |
| :<math>a \lor 0 = a</math>,
| |
| :<math>a \land 1 = a</math>.
| |
| | |
| See [[semilattice]] for further details.
| |
| | |
| ===Connection to other algebraic structures===
| |
| Lattices have some connections to the family of [[magma (algebra)|group-like algebraic structures]]. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative [[semigroups]] having the same domain. For a bounded lattice, these semigroups are in fact commutative [[monoid]]s. The [[absorption law]] is the only defining identity that is peculiar to lattice theory.
| |
| | |
| By commutativity and associativity one can think of join and meet as binary operations that are defined on non-empty finite sets, rather than on elements. In a bounded lattice the empty join and the empty meet can also be defined (as 0 and 1, respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.
| |
| | |
| The algebraic interpretation of lattices plays an essential role in [[universal algebra]].
| |
| | |
| == Connection between the two definitions ==
| |
| An order-theoretic lattice gives rise to the two binary operations <math>\lor</math> and <math>\land</math>. Since the commutative, associative and absorption laws can easily be verified for these operations, they make (''L'', <math>\lor</math>, <math>\land</math>) into a lattice in the algebraic sense.
| |
| | |
| The converse is also true. Given an algebraically defined lattice (''L'', <math>\lor</math>, <math>\land</math>), one can define a partial order ≤ on ''L'' by setting
| |
| : ''a'' ≤ ''b'' if ''a'' = ''a''<math>\land</math>''b'', or
| |
| : ''a'' ≤ ''b'' if ''b'' = ''a''<math>\lor</math>''b'',
| |
| for all elements ''a'' and ''b'' from ''L''. The laws of absorption ensure that both definitions are equivalent. One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations <math>\lor</math> and <math>\land</math>.
| |
| | |
| Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.
| |
| | |
| == Examples ==
| |
| | |
| {| style="float:right"
| |
| | [[File:N-Quadrat, gedreht.svg|thumb|x150px|'''Pic.5:''' Lattice of nonnegative integer pairs, ordered componentwise.]]
| |
| |}
| |
| {| style="float:right"
| |
| | [[File:Nat num.svg|thumb|x150px|'''Pic.4:''' Lattice of positive integers, ordered by <.]]
| |
| |}
| |
| {| style="float:right"
| |
| | [[File:Lattice of partitions of an order 4 set.svg|thumb|x150px|'''Pic.3:''' Lattice of [[partition (set theory)|partition]]s of {1,2,3,4}, ordered by "''refines''".]]
| |
| |}
| |
| {| style="float:right"
| |
| | [[File:Lattice of the divisibility of 60.svg|thumb|x150px|'''Pic.2:''' Lattice of integer divisors of 60, ordered by "''divides''".]]
| |
| |}
| |
| {| style="float:right"
| |
| | [[Image:Hasse diagram of powerset of 3.svg|thumb|x150px|'''Pic.1:''' Lattice of subsets of {x,y,z}, ordered by "''is subset of''". The name "lattice" is suggested by the form of the [[Hasse diagram]] depicting it.]]
| |
| |}
| |
| {{-}}
| |
| * For any set ''A'', the collection of all subsets of ''A'' (called the [[power set]] of ''A'') can be ordered via [[subset|subset inclusion]] to obtain a lattice bounded by ''A'' itself and the null set. Set [[intersection (set theory)|intersection]] and [[union (set theory)|union]] interpret meet and join, respectively (see pic.1).
| |
| * For any set ''A'', the collection of all finite subsets of ''A'', ordered by inclusion, is also a lattice, and will be bounded if and only if ''A'' is finite.
| |
| * For any set ''A'', the collection of all [[partition of a set|partition]]s of ''A'', ordered by [[partition of a set|refinement]], is a lattice (see pic.3).
| |
| * The [[natural number|positive integers]] in their usual order form a lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see pic.4).
| |
| * The [[Cartesian square]] of the natural numbers, ordered so that (''a,b'') ≤ (''c,d'') if ''a''≤''c'' and ''b''≤''d''. The pair (0,0) is the bottom element; there is no top (see pic.5).
| |
| * The natural numbers also form a lattice under the operations of taking the [[greatest common divisor]] and [[least common multiple]], with [[divisibility]] as the order relation: ''a'' ≤ ''b'' if ''a'' divides ''b''. 1 is bottom; 0 is top. Pic.2 shows a finite sublattice.
| |
| * Every [[complete lattice]] (also see [[#Completeness|below]]) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical [[Complete_lattice#Examples|examples]].
| |
| * The set of [[compact element]]s of an [[arithmetic lattice|arithmetic]] complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property which distinguishes arithmetic lattices from [[algebraic lattice]]s, for which the compacts do only form a [[semilattice|join-semilattice]]. Both of these classes of complete lattices are studied in [[domain theory]].
| |
| | |
| Further examples of lattices are given for each of the additional properties discussed below.
| |
| <!---stop floating mode before next section--->{{clear}}
| |
| | |
| ==Counter-examples==
| |
| | |
| {| style="float:right"
| |
| | [[File:NoLatticeDiagram.svg|thumb|x150px|'''Pic.7:''' Non-lattice poset: ''b'' and ''c'' have common upper bounds ''d'', ''e'', and ''f'', but no least one.]]
| |
| |}
| |
| {| style="float:right"
| |
| | [[File:KeinVerband.svg|thumb|x150px|'''Pic.6:''' Non-lattice poset: ''c'' and ''d'' have no common upper bound.]]
| |
| |}
| |
| | |
| Most partial ordered sets are not lattices, including the following.
| |
| | |
| * A discrete poset, meaning a poset such that ''x'' ≤ ''y'' implies ''x'' = ''y'', is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.
| |
| * Although the set {1,2,3,6} partially ordered by divisibility is a lattice, the set {1,2,3} so ordered is not a lattice because the pair 2,3 lacks a join, and it lacks a meet in {2,3,6}.
| |
| * The set {1,2,3,12,18,36} partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2,3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12,18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).
| |
| | |
| <!---stop floating mode before next section--->{{clear}}
| |
| | |
| == Morphisms of lattices ==
| |
| [[File:Monotonic but nonhomomorphic map between lattices.gif|thumb|'''Pic.8:''' Monotonic map ''f'' between lattices that preserves neither joins nor meets, since ''f''(''u'')∨''f''(''v'')=''u''’∨''u''’=''u''’≠1’=''f''(1)=''f''(''u''∨''v'') and ''f''(''u'')∧''f''(''v'')=''u''’∧''u''’=''u''’≠0’=''f''(0)=''f''(''u''∧''v'').]]
| |
| The appropriate notion of a [[morphism]] between two lattices flows easily from the [[#Lattices as algebraic structures|above]] algebraic definition. Given two lattices (''L'', ∨<sub>''L''</sub>, ∧<sub>''L''</sub>) and (''M'', ∨<sub>''M''</sub>, ∧<sub>''M''</sub>), a '''lattice homomorphism''' from ''L'' to ''M'' is a function ''f'' : ''L'' → ''M'' such that for all ''a'', ''b'' ∈ ''L'':
| |
| | |
| : ''f''(''a''∨<sub>''L''</sub>''b'') = ''f''(''a'') ∨<sub>''M''</sub> ''f''(''b''), and
| |
| : ''f''(''a''∧<sub>''L''</sub>''b'') = ''f''(''a'') ∧<sub>''M''</sub> ''f''(''b'').
| |
| | |
| Thus ''f'' is a [[homomorphism]] of the two underlying [[semilattice]]s. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a '''bounded-lattice homomorphism''' (usually called just "lattice homomorphism") ''f'' between two bounded lattices ''L'' and ''M'' should also have the following property:
| |
| | |
| : ''f''(0<sub>''L''</sub>) = 0<sub>''M''</sub> , and
| |
| : ''f''(1<sub>''L''</sub>) = 1<sub>''M''</sub> .
| |
| | |
| In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function [[limit preserving function (order theory)|preserving]] binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
| |
| | |
| Any homomorphism of lattices is necessarily [[monotone function|monotone]] with respect to the associated ordering relation; see [[limit preserving function (order theory)|preservation of limits]]. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see pic.8), although an [[Monotonic function|order-preserving]] [[bijection]] is a homomorphism if its [[inverse function|inverse]] is also order-preserving.
| |
| | |
| Given the standard definition of [[isomorphism]]s as invertible morphisms, a ''lattice isomorphism'' is just a [[bijective]] lattice homomorphism. Similarly, a ''lattice endomorphism'' is a lattice homomorphism from a lattice to itself, and a ''lattice automorphism'' is a bijective lattice endomorphism. Lattices and their homomorphisms form a [[category theory|category]].
| |
| | |
| == Sublattices ==
| |
| A ''sublattice'' of a lattice ''L'' is a nonempty subset of ''L'' that is a lattice with the same meet and join operations as ''L''. That is, if ''L'' is a lattice and ''M''<math>\not=\varnothing</math> is a subset of ''L'' such that for every pair of elements ''a'', ''b'' in ''M'' both ''a''<math>\wedge</math>''b'' and ''a''<math>\vee</math>''b'' are in ''M'', then ''M'' is a sublattice of ''L''.<ref>Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2.</ref>
| |
| | |
| A sublattice ''M'' of a lattice ''L'' is a ''convex sublattice'' of ''L'', if ''x ≤ z ≤ y'' and ''x'', ''y'' in ''M'' implies that ''z'' belongs to ''M'', for all elements ''x, y, z'' in ''L''.
| |
| | |
| == Properties of lattices ==
| |
| {{see|Map of lattices}}
| |
| We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.
| |
| | |
| === Completeness ===
| |
| {{main|Complete lattice}}
| |
| A poset is called a '''complete lattice''' if ''all'' its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.
| |
| | |
| Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.
| |
| | |
| Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
| |
| | |
| === Conditional completeness ===
| |
| A '''conditionally complete lattice''' is a lattice in which every ''nonempty'' subset ''that has an upper bound'' has a join (i.e., a least upper bound). Such lattices provide the most direct generalization of the [[completeness axiom]] of the [[real number]]s. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element 1, its minimum element 0, or both.
| |
| | |
| === Distributivity ===
| |
| {| style="float:right"
| |
| |-
| |
| | [[File:N 5 mit Beschriftung.svg|thumb|x150px|'''Pic.10:''' Smallest non-modular (and hence non-distributive) lattice N<sub>5</sub>. <BR>The labelled elements violate the distributivity equation ''c''∧(''a''∨''b'')=(''c''∧''a'')∨(''c''∧''b''), but satisfy its dual ''c''∨(''a''∧''b'')=(''c''∨''a'')∧(''c''∨''b'').]]
| |
| |}
| |
| {| style="float:right"
| |
| |-
| |
| | [[File:M 3 mit Beschriftung.svg|thumb|x150px|'''Pic.9:''' Smallest non-distributive (but modular) lattice M<sub>3</sub>.]]
| |
| |}
| |
| {{main|Distributive lattice}}
| |
| Since lattices come with two binary operations, it is natural to ask whether one of them [[distributivity|distributes]] over the other, i.e. whether one or the other of the following [[duality (order theory)|dual]] laws holds for every three elements ''a, b, c'' of ''L'':
| |
| | |
| ;Distributivity of ∨ over ∧
| |
| : ''a''∨(''b''∧''c'') = (''a''∨''b'') ∧ (''a''∨''c'').
| |
| ;Distributivity of ∧ over ∨
| |
| : ''a''∧(''b''∨''c'') = (''a''∧''b'') ∨ (''a''∧''c'').
| |
| | |
| A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a '''distributive lattice'''.
| |
| The only non-distribute lattices with fewer than 6 elements are called M<sub>3</sub> and N<sub>5</sub>,<ref>(Davey, Priestley, 2002), Exercise 6.4{{clarify|reason=Reference taken from the 1990 edition. Somebody please check whether it is different in the 2002 edition. Similar for a few subsequent marked references.|date=September 2013}}</ref> they are shown in picture 9 and 10, respectively. A lattice is distributive if and only if it doesn't have a [[#Sublattices|sublattice]] isomorphic to M<sub>3</sub> or N<sub>5</sub>.<ref name="Davey.Priestley.2002.10.6">(Davey, Priestley, 2002), Theorem 6.10{{clarify|reason=1990 reference|date=September 2013}}</ref> Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).<ref>(Davey, Priestley, 2002), Theorem 10.3{{clarify|reason=1990 reference|date=September 2013}}</ref>
| |
| | |
| For an overview of stronger notions of distributivity which are appropriate for complete lattices and which are used to define more special classes of lattices such as [[complete Heyting algebra|frames]] and [[completely distributive lattice]]s, see [[distributivity (order theory)|distributivity in order theory]].
| |
| | |
| ===Modularity===
| |
| {{main|Modular lattice}}
| |
| For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice (''L'', ∨, ∧) is '''modular''' if, for all elements ''a, b, c'' of ''L'', the following identity holds.
| |
| ;'''Modular identity''': (''a'' ∧ ''c'') ∨ (''b'' ∧ ''c'') = [(''a'' ∧ ''c'') ∨ ''b''] ∧ ''c''.
| |
| This condition is equivalent to the following axiom.
| |
| ;'''Modular law''': ''a'' ≤ ''c'' implies ''a'' ∨ (''b'' ∧ ''c'') = (''a'' ∨ ''b'') ∧ ''c''.
| |
| A lattice is modular if and only if it doesn't have a [[#Sublattices|sublattice]] isomorphic to N<sub>5</sub> (shown in pic.10).<ref name="Davey.Priestley.2002.10.6"/>
| |
| Besides distributive lattices, examples of modular lattices are the lattice of [[two-sided ideal]]s of a [[ring (mathematics)|ring]], the lattice of submodules of a [[module (mathematics)|module]], and the lattice of [[normal subgroup]]s of a [[group (mathematics)|group]]. The [[Subsumption lattice|set of first-order terms]] with the ordering "''is more specific than''" is a non-modular lattice used in [[automated reasoning]].
| |
| | |
| ===Semimodularity===
| |
| {{main|Semimodular lattice}}
| |
| A finite lattice is modular if and only if it is both upper and lower [[semimodular lattice|semimodular]]. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function ''r'':
| |
| :<math>r(x)+r(y) \ge r(x \wedge y) + r(x \vee y).</math>
| |
| Another equivalent (for graded lattices) condition is Birkhoff's condition:
| |
| : for each ''x'' and ''y'' in ''L'', if ''x'' and ''y'' both cover <math>x \wedge y</math>, then <math>x \vee y</math> covers both ''x'' and ''y''.
| |
| | |
| A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with <math>\vee</math> and <math>\wedge</math> exchanged, "covers" exchanged with "is covered by", and inequalities reversed.<ref>{{Citation | last=Stanley | first=Richard P | authorlink=Richard P. Stanley | title=Enumerative Combinatorics (vol. 1) | publisher=Cambridge University Press | pages=103–104 | isbn=0-521-66351-2}}</ref>
| |
| | |
| === Continuity and algebraicity ===
| |
| | |
| In [[domain theory]], it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of [[continuous poset]]s, consisting of posets where every element can be obtained as the supremum of a [[directed set]] of elements that are [[way-below relation|way-below]] the element. If one can additionally restrict these to the [[compact element]]s of a poset for obtaining these directed sets, then the poset is even [[algebraic poset|algebraic]]. Both concepts can be applied to lattices as follows:
| |
| | |
| * A '''continuous lattice''' is a complete lattice that is continuous as a poset.
| |
| * An '''[[algebraic lattice]]''' is a complete lattice that is algebraic as a poset.
| |
| | |
| Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via [[Scott information system]]s.
| |
| | |
| === Complements and pseudo-complements ===
| |
| Let ''L'' be a bounded lattice with greatest element 1 and least element 0. Two elements ''x'' and ''y'' of ''L'' are '''complements''' of each other if and only if:
| |
| | |
| : <math>x \vee y = 1</math> and <math>x \wedge y = 0.</math>
| |
| | |
| In the case the complement is unique, we write ¬''x'' = ''y'' and equivalently, ¬''y'' = ''x''. A bounded lattice for which every element has a complement is called a [[complemented lattice]]. The corresponding unary [[Operation (mathematics)|operation]] over ''L'', called complementation, introduces an analogue of logical [[negation]] into lattice theory. The complement is not necessarily unique, nor does it have a special status among all possible unary operations over ''L''. A complemented lattice that is also distributive is a [[Boolean algebra (structure)|Boolean algebra]]. For a distributive lattice, the complement of ''x'', when it exists, is unique.
| |
| | |
| [[Heyting algebra]]s are an example of distributive lattices where some members might be lacking complements. Every element ''x'' of a Heyting algebra has, on the other hand, a ''pseudo-complement'', also denoted ¬''x''. The pseudo-complement is the greatest element ''y'' such that ''x''<math>\wedge</math>''y'' = 0. If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.
| |
| | |
| === Jordan–Dedekind chain condition ===
| |
| A '''chain''' from ''x''<sub>0</sub> to ''x''<sub>''n''</sub> is a set <math>\{ x_0, x_1, \ldots, x_n\}</math>, where <math>x_0 < x_1 < x_2 < \ldots < x_n</math>.
| |
| The '''length''' of this chain is ''n'', or one less than its number of elements. A chain is '''maximal''' if ''x''<sub>''i''</sub> covers ''x''<sub>''i''-1</sub>
| |
| for all 1 ≤ ''i'' ≤ ''n''.
| |
| | |
| If for any pair, ''x'' and ''y'', where ''x'' < ''y'', all maximal chains from ''x'' to ''y'' have the same length, then the lattice is said to satisfy the '''Jordan–Dedekind chain condition'''.
| |
| | |
| == Free lattices ==
| |
| {{main|Free lattice}}
| |
| Any set ''X'' may be used to generate the '''free semilattice''' ''FX''. The free semilattice is defined to consist of all of the finite subsets of ''X'', with the semilattice operation given by ordinary [[set union]]. The free semilattice has the [[universal property]].
| |
| | |
| == Important lattice-theoretic notions ==
| |
| We now define some order-theoretic notions of importance to lattice theory. In the following, let ''x'' be an element of some lattice ''L''. If ''L'' has a bottom element 0, ''x''≠0 is sometimes required. ''x'' is called:
| |
| *'''Join irreducible''' if ''x'' = ''a''∨''b'' implies ''x'' = ''a'' or ''x'' = ''b'' for all ''a'', ''b'' in ''L''. When the first condition is generalized to arbitrary joins <math>\bigvee_{i \in I} a_i</math>, ''x'' is called '''completely join irreducible''' (or ∨-irreducible). The dual notion is '''meet irreducibility''' (∧-irreducible). For example, in pic.2, the elements 2, 3, 4, and 5 are join irreducible, while 12, 15, 20, and 30 are meet irreducible. In the lattice of [[real numbers]] with the usual order, each element is join irreducible, but none is completely join irreducible.
| |
| *'''Join prime''' if ''x'' ≤ ''a'' ∨ ''b'' implies ''x'' ≤ ''a'' or ''x'' ≤ ''b''. This too can be generalized to obtain the notion '''completely join prime'''. The dual notion is '''meet prime'''. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible. The converse holds if ''L'' is distributive.
| |
| | |
| Let ''L'' have a bottom element 0. An element ''x'' of ''L'' is an [[atom (order theory)|atom]] if 0 < ''x'' and there exists no element ''y'' of ''L'' such that 0 < ''y'' < ''x''. Then ''L'' is called:
| |
| * [[atomic (order theory)|Atomic]] if for every nonzero element ''x'' of ''L'', there exists an atom ''a'' of ''L'' such that ''a'' ≤ ''x'' ;
| |
| * [[atomic (order theory)|Atomistic]] if every element of ''L'' is a [[supremum]] of atoms. That is, for all ''a'', ''b'' in ''L'' such that <math>a\nleq b,</math> there exists an atom ''x'' of ''L'' such that <math>x\leq a</math> and <math>x\nleq b.</math>
| |
| | |
| The notions of [[ideal (order theory)|ideal]]s and the dual notion of [[filter (mathematics)|filters]] refer to particular kinds of [[subset]]s of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
| |
| | |
| == See also ==
| |
| {{columns-start|num=3}}
| |
| *[[Join and meet]]
| |
| *[[Map of lattices]]
| |
| *[[Orthocomplemented lattice]]
| |
| *[[Total order]]
| |
| *[[Ideal (order theory)|Ideal]] and [[Filter (mathematics)|Filter]] (dual notions)
| |
| *[[Skew lattice]] (generalization to non-commutative join and meet)
| |
| *[[Eulerian lattice]]
| |
| *[[Post's lattice]]
| |
| *[[Tamari lattice]]
| |
| *[[Young–Fibonacci lattice]]
| |
| {{column}}
| |
| | |
| ===Applications that use lattice theory===
| |
| ''Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.''
| |
| *[[Pointless topology]]
| |
| *[[Lattice of subgroups]]
| |
| *[[Spectral space]]
| |
| *[[Invariant subspace]]
| |
| *[[Closure operator]]
| |
| *[[Abstract interpretation]]
| |
| *[[Subsumption lattice]]
| |
| *[[Fuzzy set]] theory
| |
| {{column}}
| |
| *[[First-order_logic#Algebraizations|Algebraizations of first-order logic]]
| |
| *[[Ontology (computer science)]]
| |
| *[[Multiple inheritance]]
| |
| *[[Formal concept analysis]] and [[Lattice Miner]] (theory and tool)
| |
| *[[Bloom filter#Compact approximators|Bloom filter]]
| |
| *[[Information flow]]
| |
| *[[Ordinal optimization]]
| |
| *[[Quantum logic]]
| |
| *[[Median graph]]
| |
| *[[Knowledge space]]
| |
| {{columns-end}}
| |
| | |
| == Notes ==
| |
| {{reflist|group=note}}
| |
| | |
| == References ==
| |
| | |
| {{reflist}}
| |
| | |
| Monographs available free online:
| |
| * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2.
| |
| * Jipsen, Peter, and Henry Rose, ''[http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html Varieties of Lattices]'', Lecture Notes in Mathematics 1533, Springer Verlag, 1992. ISBN 0-387-56314-8.
| |
| *Nation, J. B., ''Notes on Lattice Theory''. [http://www.math.hawaii.edu/~jb/lat1-6.pdf Chapters 1-6.] [http://www.math.hawaii.edu/~jb/lat7-12.pdf Chapters 7-12; Appendices 1-3.]
| |
| | |
| Elementary texts recommended for those with limited [[mathematical maturity]]:
| |
| *Donnellan, Thomas, 1968. ''Lattice Theory''. Pergamon.
| |
| *Grätzer, G., 1971. ''Lattice Theory: First concepts and distributive lattices''. W. H. Freeman.
| |
| | |
| The standard contemporary introductory text, somewhat harder than the above:
| |
| * {{Citation | last1=Davey | first1=B.A. | last2=Priestley | first2=H. A. | title=Introduction to Lattices and Order | publisher=[[Cambridge University Press]] | isbn=978-0-521-78451-1 | year=2002}}
| |
| | |
| Advanced monographs:
| |
| * [[Garrett Birkhoff]], 1967. ''Lattice Theory'', 3rd ed. Vol. 25 of AMS Colloquium Publications. [[American Mathematical Society]].
| |
| *[[Robert P. Dilworth]] and Crawley, Peter, 1973. ''Algebraic Theory of Lattices''. Prentice-Hall. ISBN 978-0-13-022269-5.
| |
| *{{cite isbn|9783764369965}}
| |
| | |
| On free lattices:
| |
| * R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42. [[Mathematical Association of America]].
| |
| * Johnstone, P.T., 1982. ''Stone spaces''. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.
| |
| | |
| On the history of lattice theory:
| |
| * {{cite book| author=Štĕpánka Bilová| title=Lattice theory — its birth and life| year=2001| pages=250–257| publisher=Prometheus| editor=Eduard Fuchs| url=http://dml.cz/bitstream/handle/10338.dmlcz/401261/DejinyMat_17-2001-1_31.pdf}}
| |
| | |
| On applications of lattice theory:
| |
| * {{cite book| author=Garrett Birkhoff| title=What can Lattices do for you?| year=1967| publisher=Van Nostrand| editor=James C. Abbot }} [http://www.ulb.tu-darmstadt.de/tocs/129983330.pdf Table of contents]
| |
| | |
| ==External links==
| |
| {{Commons|Lattice (order)}}
| |
| * {{springer|title=Lattice-ordered group|id=p/l057670}}
| |
| * {{Mathworld|urlname=Lattice |title=Lattice}}
| |
| * J.B. Nation, [http://www.math.hawaii.edu/~jb/books.html ''Notes on Lattice Theory''], unpublished course notes available as two PDF files.
| |
| * Ralph Freese, "[http://www.math.hawaii.edu/LatThy/ Lattice Theory Homepage]".
| |
| | |
| [[Category:Lattice theory| ]]
| |
| [[Category:Algebraic structures]]
| |