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In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all ''x'' in a semigroup ''S'', there exists a positive integer ''n'' and a subgroup ''G'' of ''S'' such that ''x''''n'' belongs to ''G''. Epigroups are known by wide variety of other names, including quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr),〔A. V. Kelarev, ''Applications of epigroups to graded ring theory'', Semigroup Forum, Volume 50, Number 1 (1995), 327–350 〕 or just π-regular semigroup (although the latter is ambiguous). More generally, in an arbitrary semigroup an element is called ''group-bound'' if it has a power that belongs to a subgroup. Epigroups have applications to ring theory. Many of their properties are studied in this context. Epigroups were fist studied by Douglas Munn in 1961, who called them ''pseudoinvertible''. == Properties == * Epigroups are a generalization of periodic semigroups, thus all finite semigroups are also epigroups. * The class of epigroups also contains all completely regular semigroups and all completely 0-simple semigroups.〔 * All epigroups are also eventually regular semigroups. (also known as π-regular semigroups) * A cancellative epigroup is a group. * Green's relations ''D'' and ''J'' coincide for any epigroup. * If ''S'' is an epigroup, any regular subsemigroup of ''S'' is also an epigroup.〔 * In an epigroup the Nambooripad order (as extended by P.R. Jones) and the natural partial order (of Mitsch) 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「Epigroup」の詳細全文を読む スポンサード リンク
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