:Null semigroup#Left zero element and right zero element in a semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero.{{cite book| last=A H Clifford|author2=G B Preston |title=The Algebraic Theory of Semigroups, volume I|publisher=American Mathematical Society| date=1964|edition=2|series=mathematical Surveys|volume=1|pages=3–4|isbn=978-0-8218-0272-4}} If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, {{isbn|3-11-015248-7}}, p. 19
According to A. H. Clifford and G. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."
Null semigroup
Let S be a semigroup with zero element 0. Then S is called a null semigroup if xy = 0 for all x and y in S.
=Cayley table for a null semigroup=
Let S = {0, a, b, c} be (the underlying set of) a null semigroup. Then the Cayley table for S is as given below:
| class="wikitable" style="width: 25%"
|+Cayley table for a null semigroup |+ ! ! 0 !a !b !c |
| 0
| 0 | 0 | 0 | 0 |
|---|
| a
| 0 | 0 | 0 | 0 |
| b
| 0 | 0 | 0 | 0 |
| c
| 0 | 0 | 0 | 0 |
Left zero semigroup
A semigroup in which every element is a left zero element is called a left zero semigroup. Thus a semigroup S is a left zero semigroup if xy = x for all x and y in S.
=Cayley table for a left zero semigroup=
Let S = {a, b, c} be a left zero semigroup. Then the Cayley table for S is as given below:
| class="wikitable" style="width: 25%"
|+Cayley table for a left zero semigroup |+ ! !a !b !c |
| a
| a | a | a |
|---|
| b
| b | b | b |
| c
| c | c | c |
Right zero semigroup
A semigroup in which every element is a right zero element is called a right zero semigroup. Thus a semigroup S is a right zero semigroup if xy = y for all x and y in S.
=Cayley table for a right zero semigroup=
Let S = {a, b, c} be a right zero semigroup. Then the Cayley table for S is as given below:
| class="wikitable" style="width: 25%"
|+Cayley table for a right zero semigroup |+ ! ! a !b !c |
| a
| a | b | c |
|---|
| b
| a | b | c |
| c
| a | b | c |
Properties
A non-trivial null (left/right zero) semigroup does not contain an identity element. It follows that the only null (left/right zero) monoid is the trivial monoid.
The class of null semigroups is:
- closed under taking subsemigroups
- closed under taking quotient of subsemigroup
- closed under arbitrary direct products.
It follows that the class of null (left/right zero) semigroups is a variety of universal algebra, and thus a variety of finite semigroups. The variety of finite null semigroups is defined by the identity ab = cd.
See also
References
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