Trivial semigroup

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 = { a } is a semigroup with one element, then the Cayley table of S is

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The only element in S is the zero element 0 of S and is also the identity element 1 of S.{{cite book |author=A. H. Clifford |author2=G. B. Preston |year=1964 |title=The Algebraic Theory of Semigroups |volume=I |edition=2nd |publisher=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.{{cite book |author=P. A. Grillet |year=1995 |title=Semigroups |publisher=CRC Press |ISBN=978-0-8247-9662-4 |pages=3–4}}{{cite book |first = J. M.|last = Howie |author-link = John Mackintosh Howie|title=An Introduction to Semigroup Theory |publisher=Academic Press |year=1976 |series=LMS Monographs |volume=7 |pages=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.