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In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one. If ''S'' = is a semigroup with one element then the Cayley table of ''S'' is as given below: The only element in ''S'' is the zero element 0 of ''S'' and is also the identity element 1 of ''S''.〔A H Clifford, G B Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). American Mathematical Society. ISBN 978-0-8218-0272-4〕 However not all semigroup theorists consider the unique element in a semigroup with one element as the zero element of the semigroup. They define zero elements only in semigroups having at least two elements.〔 P A Grillet (1995). ''Semigroups''. CRC Press. ISBN 978-0-8247-9662-4 pp.3-4〕〔 pp.2-3〕 In spite of its extreme triviality, the semigroup with one element is important in many situations. It is the starting point for understanding the structure of semigroups. It serves as a counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal. Further, if ''S'' is a semigroup with one element, the semigroup obtained by adjoining an identity element to ''S'' is isomorphic to the semigroup obtained by adjoining a zero element to ''S''. The semigroup with one element is also a group. In the language of category theory, any semigroup with one element is a terminal object in the category of semigroups. ==See also== * Field with one element * Empty semigroup * Semigroup with two elements * Special classes of semigroups 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「Trivial semigroup」の詳細全文を読む スポンサード リンク
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