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In abstract algebra, in semigroup theory, a Schutzenberger group is a certain group associated with a Green ''H''-class of a semigroup. The Schutzenberger groups associated with different ''H''-classes are different. However, the groups associated with two different ''H''-classes contained in the same ''D''-class of a semigroup are isomorphic. Moreover, if the ''H''-class itself were a group, the Schutzenberger group of the ''H''-class would be isomorphic to the ''H''-class. In fact, there are two Schutzenberger groups associated with a given ''H''-class and each is antiisomorphic to the other. The Schutzenberger group was discovered by Marcel-Paul Schützenberger in 1957〔 (MR 19, 249)〕〔 (pp. 63–66)〕 and the terminology was coined by A. H. Clifford. ==The Schutzenberger group== Let ''S'' be a semigroup and let ''S''1 be the semigroup obtained by adjoining an identity element 1 to ''S'' (if ''S'' already has an identity element, then ''S''1 = ''S''). Green's ''H''-relation in ''S'' is defined as follows: If ''a'' and ''b'' are in ''S'' then :''a'' ''H'' ''b'' ⇔ there are ''u'', ''v'', ''x'', ''y'' in ''S''1 such that ''ua'' = ''ax'' = ''b'' and ''vb'' = ''by'' = ''a''. For ''a'' in ''S'', the set of all ''b'' 's in ''S'' such that ' ''a'' ''H'' ''b'' ' is the Green ''H''-class of ''S'' containing ''a'', denoted by ''H''''a''. Let ''H'' be an ''H''-class of the semigroup ''S''. Let ''T''( ''H'' ) be the set of all elements ''t'' in ''S''1 such that ''Ht'' is a subset of ''H'' itself. Each ''t'' in ''T''( ''H'' ) defines a transformation, denoted by γ''t'', of ''H'' by mapping ''h'' in ''H'' to ''ht'' in ''H''. The set of all these transformations of ''H'', denoted by Γ ( ''H'' ), is a group under composition of mappings (taking functions as right operators). The group Γ ( ''H'' ) is the Schutzenberger group associated with the ''H''-class ''H''. 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「Schutzenberger group」の詳細全文を読む スポンサード リンク
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