|
|
| Line 1: |
Line 1: |
| In [[algebra]] and [[theoretical computer science]], an '''action''' or '''act''' of a '''[[semigroup]]''' on a [[Set (mathematics)|set]] is a rule which associates to each element of the semigroup a [[transformation (geometry)|transformation]] of the set in such a way that the product of two elements of the semigroup (using the semigroup [[binary operation|operation]]) is associated with the [[function composition|composite]] of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are ''acting'' as transformations of the set. From an [[algebraic structure|algebraic]] perspective, a semigroup action is a generalization of the notion of a [[group action]] in [[group (mathematics)|group theory]]. From the computer science point of view, semigroup actions are closely related to [[finite state machine|automata]]: the set models the state of the automaton and the action models transformations of that state in response to inputs.
| | Hello and welcome. My name is Numbers Wunder. To play baseball is the pastime he will by no means stop performing. California is exactly where I've usually been living and I love every day residing right here. For many years he's been operating as a meter reader and it's some thing he really enjoy.<br><br>my page [http://bonetoob.com/blog/381389 home std test] |
| | |
| An important special case is a '''monoid action''' or '''act''', in which the semigroup is a [[monoid]] and the [[identity element]] of the monoid acts as the [[identity transformation]] of a set. From a [[category theoretic]] point of view, a monoid is a [[category (mathematics)|category]] with one object, and an act is a functor from that category to the [[category of sets]]. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets.
| |
| | |
| Another important special case is a '''[[transformation semigroup]]'''. This is a semigroup of transformations of a set, and hence it has a tautological action of on that set. This concept is linked to the more general notion of a semigroup by an analogue of [[Cayley's theorem]].
| |
| | |
| ''(A note on terminology: the terminology used in this area varies, sometimes significantly, from one author to another. See the article for details.)''
| |
| | |
| ==Formal definitions==
| |
| | |
| Let ''S'' be a semigroup. Then a (left) '''semigroup action''' (or '''act''') of ''S'' is a set ''X'' together with an operation {{nowrap|• : ''S'' × ''X'' → ''X''}} which is compatible with the semigroup [[binary operation|operation]] * as follows:
| |
| * for all ''s'', ''t'' in ''S'' and ''x'' in ''X'', {{nowrap|1=''s'' • (''t'' • ''x'') = (''s'' * ''t'') • ''x''}}.
| |
| This is the analogue in semigroup theory of a (left) [[group action]], and is equivalent to a [[semigroup#Semigroup homomorphisms|semigroup homomorphism]] into the set of functions on ''X''. Right semigroup actions are defined in a similar way using an operation {{nowrap|• : ''X'' × ''S'' → ''X''}} satisfying {{nowrap|1=(''x'' • ''a'') • ''b'' = ''x'' • (''a'' * ''b'')}}.
| |
| | |
| If ''M'' is a monoid, then a (left) '''monoid action''' (or '''act''') of ''M'' is a (left) semigroup action of ''M'' with the additional property that
| |
| * for all ''x'' in ''X'': ''e'' • ''x'' = ''x''
| |
| where ''e'' is the identity element of ''M''. This correspondingly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid ''M'' with an action on a set is also called an '''operator monoid'''.
| |
| | |
| A semigroup action of ''S'' on ''X'' can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on ''X''.
| |
| | |
| ===Terminology and notation===
| |
| | |
| If ''S'' is a semigroup or monoid, then a set ''X'' on which ''S'' acts as above (on the left, say) is also known as a (left) '''''S''-act''', '''''S''-set''', '''''S''-action''', '''''S''-operand''', or '''left act over ''S'''''. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom ({{nowrap|1=''e'' • ''x'' = ''x''}}) as empty when there is no identity element, or by using the term '''unitary ''S''-act''' for an ''S''-act with an identity.<ref>Kilp, Knauer and Mikhalev, 2000, pages 43–44.</ref> Furthermore, since a monoid is a semigroup, one can consider semigroup actions of monoids.
| |
| | |
| The defining property of an act is analogous to the [[associativity]] of the semigroup operation, and means that all parentheses can be omitted. It is common practice, especially in computer science, to omit the operations as well so that both the semigroup operation and the action are indicated by juxtaposition. In this way [[string (computer science)|strings]] of letters from ''S'' act on ''X'', as in the expression ''stx'' for ''s'', ''t'' in ''S'' and ''x'' in ''X''.
| |
| | |
| It is also quite common to work with right acts rather than left acts.<ref>Kilp, Knauer and Mikhalev, 2000.</ref> However, every right S-act can be interpreted as a left act over the '''opposite monoid''', which has the same elements as S, but where multiplication is defined by reversing the factors, {{nowrap|1=''s'' • ''t'' = ''t'' • ''s''}}, so the two notions are essentially equivalent. Here we primarily adopt the point of view of left acts.
| |
| | |
| ===Acts and transformations===
| |
| | |
| It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as <math>T</math>, to denote the function
| |
| :<math> T\colon S\times X \to X</math>
| |
| defining the <math>S</math>-action and hence write <math>T(s, x)</math> in place of <math>s\cdot x</math>. Then for any <math>s</math> in <math>S</math>, we denote by
| |
| :<math> T_s\colon X \to X</math>
| |
| the transformation of <math>X</math> defined by
| |
| :<math> T_s(x) = T(s,x).</math>
| |
| By the defining property of an <math>S</math>-act, <math>T</math> satisfies
| |
| :<math> T_{s*t} = T_s\circ T_t.</math>
| |
| Further, consider a function <math>s\mapsto T_s</math>. It is the same as <math>curry(T):S\to(X\to X)</math> (see [[currying]]). Because <math>curry</math> is a bijection, semigroup actions can be defined as functions <math>S\to(X\to X)</math> which satisfies
| |
| :<math> curry(T)(s*t) = curry(T)(s)\circ curry(T)(t).</math>
| |
| I.e. <math>T</math> is a semigroup action of <math>S</math> on <math>X</math> iff <math>curry(T)</math> is a [[semigroup homomorphism]] from <math>S</math> to the full transformation monoid of <math>X</math>.
| |
| | |
| ===''S''-homomorphisms===
| |
| | |
| Let ''X'' and ''X''′ be ''S''-acts. Then an ''S''-homomorphism from ''X'' to ''X''′ is a map
| |
| :<math>F\colon X\to X'</math>
| |
| such that
| |
| :<math>F(sx) =s F(x)</math> for all <math>s\in S</math> and <math>x\in X</math>.
| |
| The set of all such ''S''-homomorphisms is commonly written as <math>\mathrm{Hom}_S(X,X')</math>.
| |
| | |
| ''M''-homomorphisms of ''M''-acts, for ''M'' a monoid, are defined in exactly the same way.
| |
| | |
| ===''S''-Act and ''M''-Act===
| |
| | |
| For a fixed semigroup ''S'', the left ''S''-acts are the objects of a category, denoted ''S''-Act, whose morphisms are the ''S''-homomorphisms. The corresponding category of right ''S''-acts is sometimes denoted by Act-''S''. (This is analogous to the categories ''R''-Mod and Mod-''R'' of left and right [[module (mathematics)|modules]] over a [[ring (mathematics)|ring]].)
| |
| | |
| For a monoid ''M'', the categories ''M''-Act and Act-''M'' are defined in the same way.
| |
| | |
| ==Transformation semigroups==
| |
| | |
| {{main|Transformation semigroup}}
| |
| | |
| A correspondence between transformation semigroups and semigroup actions is described below. If we restrict it to [[Faithful action|faithful]] semigroup actions, it has nice properties.
| |
| | |
| Any transformation semigroup can be turned into a semigroup action by the following construction. For any transformation semigroup <math>S</math> of <math>X</math>, define a semigroup action <math>T</math> of <math>S</math> on <math>X</math> as <math>T(s, x) = s(x)</math> for <math> s\in S, x\in X</math>. This action is faithful, which is equivalent to <math>curry(T)</math> being [[injective]].
| |
| | |
| Conversely, for any semigroup action <math>T</math> of <math>S</math> on <math>X</math>, define a transformation semigroup <math>S' = \{T_s | s\in S\}</math>. In this construction we "forget" the set <math>S</math>. <math>S'</math> is equal to the [[Image (mathematics)|image]] of <math>curry(T)</math>. Lets denote <math>curry(T)</math> as <math>f</math> for brevity. If <math>curry(T)</math> is [[injective]], then <math>f</math> is a semigroup [[isomorphism]] from <math>S</math> to <math>S'</math>. In other words, if <math>T</math> is faithful, then we forget nothing important. This claim is made precise by the following observation: if we turn <math>S'</math> back into a semigroup action <math>T'</math> of <math>S'</math> on <math>X</math>, then <math>T'(f(s), x)=T(s, x)</math> for all <math> s\in S, x\in X</math>. <math>T</math> and <math>T'</math> are "isomorphic" via <math>f</math>, i.e., we essentially recovered <math>T</math>. Thus some authors<ref>{{cite book
| |
| | editor1-first = Michael A.
| |
| | editor1-last = Arbib
| |
| | year = 1968
| |
| | title = Algebraic Theory of Machines, Languages, and Semigroups
| |
| | publisher = Academic Press
| |
| | location = New York and London
| |
| | page = 83
| |
| }}</ref> see no distinction between faithful semigroup actions and transformation semigroups.
| |
| | |
| ==Applications to computer science==
| |
| ===Semiautomata===
| |
| | |
| {{main|semiautomaton}}
| |
| | |
| Transformation semigroups are of essential importance for the structure theory of [[finite state machine]]s in [[automata theory]]. In particular, a ''semiautomaton'' is a triple (''Σ'',''X'',''T''), where ''Σ'' is a non-empty set called the ''input alphabet'', ''X'' is a non-empty set called the ''set of states'' and ''T'' is a function
| |
| :<math>T\colon \Sigma\times X \to X</math> | |
| called the ''transition function''. Semiautomata arise from [[deterministic finite automaton|deterministic automata]] by ignoring the initial state and the set of accept states.
| |
| | |
| Given a semiautomaton, let ''T''<sub>''a''</sub>: ''X'' → ''X'', for ''a'' ∈ ''Σ'', denote the transformation of ''X'' defined by ''T''<sub>''a''</sub>(''x'') = ''T''(''a'',''x''). Then semigroup of transformations of ''X'' generated by {''T''<sub>''a''</sub> : ''a'' ∈ ''Σ''} is called the ''[[characteristic semigroup]]'' or ''transition system'' of (''Σ'',''X'',''T''). This semigroup is a monoid, so this monoid is called the ''characteristic'' or ''[[transition monoid]]''. It is also sometimes viewed as an ''Σ''<sup>∗</sup>-act on ''X'', where ''Σ''<sup>∗</sup> is the [[free monoid]] of strings generated by the alphabet ''Σ'' and the action of strings extends the action of ''Σ'' via the property
| |
| :<math>T_{vw} = T_w \circ T_v.</math>
| |
| | |
| ===Krohn–Rhodes theory===
| |
| | |
| {{main|Krohn–Rhodes theory}}
| |
| | |
| Krohn–Rhodes theory, sometimes also called ''algebraic automata theory'', gives powerful decomposition results for finite transformation semigroups by cascading simpler components.
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| * A. H. Clifford and G. B. Preston (1961), ''The Algebraic Theory of Semigroups'', volume 1. American Mathematical Society, ISBN 978-0-8218-0272-4.
| |
| * A. H. Clifford and G. B. Preston (1967), ''The Algebraic Theory of Semigroups'', volume 2. American Mathematical Society, ISBN 978-0-8218-0272-4.
| |
| * Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), ''Monoids, Acts and Categories: with Applications to Wreath Products and Graphs'', Expositions in Mathematics '''29''', Walter de Gruyter, Berlin, ISBN 978-3-11-015248-7.
| |
| * Rudolf Lidl and Günter Pilz, ''Applied Abstract Algebra'' (1998), Springer, ISBN 978-0-387-98290-8
| |
| | |
| [[Category:Semigroup theory]]
| |
| [[Category:Theoretical computer science]]
| |