Trivial semigroup: Difference between revisions
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The semigroup with one element is also a [[group]]. |
The semigroup with one element is also a [[group]]. |
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==See also== |
==See also== |
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==References== |
==References== |
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*A H Clifford, G B Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). [[American Mathematical Society]]. ISBN 978-0821802724 |
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*P A Grillet (1995). ''Semigroups''. [[CRC Press]]. ISBN 978-0824796624 |
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[[Category:Mathematics]] |
[[Category:Mathematics]] |
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Revision as of 01:07, 11 May 2009
In mathematics, 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.
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.
The semigroup with one element is also a group.
See also
References
- A H Clifford, G B Preston (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. ISBN 978-0821802724
- P A Grillet (1995). Semigroups. CRC Press. ISBN 978-0824796624