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
a a a
The only element in S is the zero element 0 of S and is also the identity element 1 of S.[1] 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.[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
- Trivial group
- Zero ring
- Field with one element
- Empty semigroup
- Semigroup with two elements
- Semigroup with three elements
- Special classes of semigroups
References
- ↑ A. H. Clifford; G. B. Preston (1964). The Algebraic Theory of Semigroups. I (2nd ed.). American Mathematical Society. ISBN 978-0-8218-0272-4.
- ↑ P. A. Grillet (1995). Semigroups. CRC Press. pp. 3–4. ISBN 978-0-8247-9662-4.
- ↑ Howie, J. M. (1976). An Introduction to Semigroup Theory. LMS Monographs. 7. Academic Press. pp. 2–3.
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