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In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup. Let ''S'' be a semigroup and ''I'' be an ideal of ''S''. Using ''S'' and ''I'' one can construct a new semigroup by collapsing ''I'' into a single element while the elements of ''S'' outside of ''I'' retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of ''S'' modulo ''I'' and is denoted by ''S''/''I''. The concept of Rees factor semigroup was introduced by David Rees in 1940.〔 MR 2, 127〕 ==Formal definition== A subset ''A'' of a semigroup ''S'' is called an ideal of ''S'' if both ''SA'' and ''AS'' are subsets of ''A''. Let ''I'' be an ideal of a semigroup ''S''. The relation ρ in ''S'' defined by : ''x'' ρ ''y'' ⇔ either ''x'' = ''y'' or both ''x'' and ''y'' are in ''I'' is an equivalence relation in ''S''. The equivalence classes under ρ are the singleton sets with ''x'' not in ''I'' and the set ''I''. Since ''I'' is an ideal of ''S'', the relation ρ is a congruence on ''S''.〔Lawson (1998), (p. 60 )〕 The quotient semigroup ''S''/ρ is, by definition, the Rees factor semigroup of ''S'' modulo ''I''. For notational convenience the semigroup ''S''/ρ is also denoted as ''S''/''I''. In the Rees factor semigroup, the product of two elements in ''S'' \ ''I'' (the complement of ''S'' and ''I'') is the same as their product in S if that product lies in ''S'' \ ''I''; if otherwise, the product is given by the new element ''I''. The congruence ρ on ''S'' as defined above is called the Rees congruence on ''S'' modulo ''I''. 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「Rees factor semigroup」の詳細全文を読む スポンサード リンク
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