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In mathematics, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique inverse ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, and composing functions from left to right — a convention often observed in semigroup theory.) == Origins == Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner〔Since his father was German, Wagner preferred the German transliteration of his name (with a "W", rather than a "V") from Cyrillic — see .〕 in the Soviet Union in 1952,〔First a short announcement in , then a much more comprehensive exposition in .〕 and by Gordon Preston in Great Britain in 1954.〔,b,c.〕 Both authors arrived at inverse semigroups via the study of partial one-one transformations of a set: a partial transformation α of a set ''X'' is a function from ''A'' to ''B'', where ''A'' and ''B'' are subsets of ''X''. Let α and β be partial transformations of a set ''X''; α and β can be composed (from left to right) on the largest domain upon which it "makes sense" to compose them: :dom αβ = (α dom β )α−1 where α−1 denotes the preimage under α. Partial transformations had already been studied in the context of pseudogroups.〔See, for example, .〕 It was Wagner, however, who was the first to observe that the composition of partial transformations is a special case of the composition of binary relations. He recognised also that the domain of composition of two partial transformations may be the empty set, so he introduced an ''empty transformation'' to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined associative binary operation. Under this composition, the collection of all partial one-one transformations of a set ''X'' forms an inverse semigroup, called the ''symmetric inverse semigroup'' (or monoid) on ''X''. This is the "archetypal" inverse semigroup, in the same way that a symmetric group is the archetypal group. For example, just as every group can be embedded in a symmetric group, every inverse semigroup can be embedded in a symmetric inverse semigroup (see below). 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「Inverse semigroup」の詳細全文を読む スポンサード リンク
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