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 as given below:

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

References

  1. ↑ 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
  2. ↑ P A Grillet (1995). Semigroups. CRC Press. ISBN 978-0-8247-9662-4 pp.3-4
  3. ↑ J M Howie (1976). An Introduction to Semigroup Theory. L.M.S.Monographs 7. Academic Press. pp.2-3