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In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility. Instead of working directly with a semigroup ''S'', it is convenient to define Green's relations over the monoid ''S''1. (''S''1 is "''S'' with an identity adjoined if necessary"; if ''S'' is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by some semigroup element do indeed contain that element. For an element ''a'' of ''S'', the relevant ideals are: * The ''principal left ideal'' generated by ''a'': . This is the same as , which is . * The ''principal right ideal'' generated by ''a'': , or equivalently . * The ''principal two-sided ideal'' generated by ''a'': , or . ==The L, R, and J relations== For elements ''a'' and ''b'' of ''S'', Green's relations ''L'', ''R'' and ''J'' are defined by * ''a'' ''L'' ''b'' if and only if ''S''1 ''a'' = ''S''1 ''b''. * ''a'' ''R'' ''b'' if and only if ''a'' ''S''1 = ''b'' ''S''1. * ''a'' ''J'' ''b'' if and only if ''S''1 ''a'' ''S''1 = ''S''1 ''b'' ''S''1. That is, ''a'' and ''b'' are ''L''-related if they generate the same left ideal; ''R''-related if they generate the same right ideal; and ''J''-related if they generate the same two-sided ideal. These are equivalence relations on ''S'', so each of them yields a partition of ''S'' into equivalence classes. The ''L''-class of ''a'' is denoted ''L''''a'' (and similarly for the other relations). Green used the lowercase blackletter , and for these relations, and wrote for ''a'' ''L'' ''b'' (and likewise for ''R'' and ''J''). Mathematicians today tend to use script letters such as instead, and replace Green's modular arithmetic-style notation with the infix style used here. Ordinary letters are used for the equivalence classes. The ''L'' and ''R'' relations are left-right dual to one another; theorems concerning one can be translated into similar statements about the other. For example, ''L'' is ''right-compatible'': if ''a'' ''L'' ''b'' and ''c'' is another element of ''S'', then ''ac'' ''L'' ''bc''. Dually, ''R'' is ''left-compatible'': if ''a'' ''R'' ''b'', then ''ca'' ''R'' ''cb''. If ''S'' is commutative, then ''L'', ''R'' and ''J'' coincide. 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「Green's relations」の詳細全文を読む スポンサード リンク
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