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__NOTOC__ In abstract algebra, an ''E''-dense semigroup (also called an ''E''-inversive semigroup) is a semigroup in which every element ''a'' has at least one weak inverse ''x'', meaning that ''xax'' = ''x''.〔 (preprint )〕 The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that ''axa''=''a''). The above definition of an ''E''-inversive semigroup ''S'' is equivalent with any of the following:〔 * for every element ''a'' ∈ ''S'' there exists another element ''b'' ∈ ''S'' such that ''ab'' is an idempotent. * for every element ''a'' ∈ ''S'' there exists another element ''c'' ∈ ''S'' such that ''ca'' is an idempotent. This explains the name of the notion as the set of idempotents of a semigroup ''S'' is typically denoted by ''E''(''S'').〔 The concept of ''E''-inversive semigroup was introduced by Gabriel Thierrin in 1955.〔Manoj Siripitukdet and Supavinee Sattayaporn (Semilattice Congruences on E-inversive Semigroups ), NU Science Journal 2007; 4(S1): 40 - 44〕〔G. Thierrin (1955), 'Demigroupes inverses et rectangularies', Bull. Cl. Sci. Acad. Roy. Belgique 41, 83-92.〕 Some authors use ''E''-dense to refer only to ''E''-inversive semigroups in which the idempotents commute. More generally, a subsemigroup ''T'' of ''S'' is said dense in ''S'' if, for all ''x'' ∈ ''S'', there exists ''y'' ∈ ''S'' such that both ''xy'' ∈ ''T'' and ''yx'' ∈ ''T''. A semigroup with zero is said to be an ''E'' *-dense semigroup if every element other than the zero has at least one non-zero weak inverse. Semigroups in this class have also been called 0-inversive semigroups.〔 (preprint )〕 == Examples == * Any regular semigroup is ''E''-dense (but not vice versa).〔 * Any eventually regular semigroup is ''E''-dense.〔 * Any periodic semigroup (and in particular, any finite semigroup) is ''E''-dense.〔 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「E-dense semigroup」の詳細全文を読む スポンサード リンク
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