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In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup anologue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being effective, i.e., if two elements of the semigroup have the same action, then they are equal. An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set. In automata theory, some authors use the term ''transformation semigroup'' to refer to a semigroup acting faithfully on a set of "states" different from the semigroup's base set. There is a correspondence between the two notions. ==Transformation semigroups and monoids== A transformation semigroup is a pair (''X'',''S''), where ''X'' is a set and ''S'' is a semigroup of transformations of ''X''. Here a transformation of ''X'' is just a function from ''X'' to itself, not necessarily invertible, and therefore ''S'' is simply a set of transformations of ''X'' which is closed under composition of functions. If ''S'' includes the identity transformation of ''X'', then it is called a transformation monoid. Obviously any transformation semigroup ''S'' determines a transformation monoid ''M'' by taking the union of ''S'' with the identity transformation. A transformation monoid whose elements are invertible is a permutation group. The set of all transformations of ''X'' is a transformation monoid called the full transformation monoid (or semigroup) of ''X''. It is also called the symmetric semigroup of ''X'' and is denoted by ''T''''X''. Thus a transformation semigroup (or monoid) is just a subsemigroup (or submonoid) of the full transformation monoid of ''X''. The full transformation monoid is a regular semigroup. If (''X'',''S'') is a transformation semigroup then ''X'' can be made into a semigroup action of ''S'' by evaluation: : This is a monoid action if ''S'' is a transformation monoid. The characteristic feature of transformation semigroups, as actions, is that they are ''effective'', i.e., if : then ''s'' = ''t''. Conversely if a semigroup ''S'' acts on a set ''X'' by ''T''(''s'',''x'') = ''s'' • ''x'' then we can define, for ''s'' ∈ ''S'', a transformation ''T''''s'' of ''X'' by : The map sending ''s'' to ''T''''s'' is injective if and only if (''X'', ''T'') is effective, in which case the image of this map is a transformation semigroup isomorphic to ''S''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Transformation semigroup」の詳細全文を読む スポンサード リンク
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