Inmathematics, anull semigroup (also called azero semigroup) is asemigroup with anabsorbing element, calledzero, in which the product of any two elements is zero.[1] If every element of a semigroup is aleft zero then the semigroup is called aleft zero semigroup; aright zero semigroup is defined analogously.[2]
According toA. H. Clifford andG. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."[1]
LetS be a semigroup with zero element 0. ThenS is called anull semigroup ifxy = 0 for allx andy inS.
LetS = {0,a,b,c} be (the underlying set of) a null semigroup. Then theCayley table forS is as given below:
| 0 | a | b | c | |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| a | 0 | 0 | 0 | 0 |
| b | 0 | 0 | 0 | 0 |
| c | 0 | 0 | 0 | 0 |
A semigroup in which every element is aleft zero element is called aleft zero semigroup. Thus a semigroupS is a left zero semigroup ifxy =x for allx andy inS.
LetS = {a,b,c} be a left zero semigroup. Then the Cayley table forS is as given below:
| a | b | c | |
|---|---|---|---|
| a | a | a | a |
| b | b | b | b |
| c | c | c | c |
A semigroup in which every element is aright zero element is called aright zero semigroup. Thus a semigroupS is a right zero semigroup ifxy =y for allx andy inS.
LetS = {a,b,c} be a right zero semigroup. Then the Cayley table forS is as given below:
| a | b | c | |
|---|---|---|---|
| a | a | b | c |
| b | a | b | c |
| c | a | b | c |
A non-trivial null (left/right zero) semigroup does not contain anidentity element. It follows that the only null (left/right zero)monoid is the trivial monoid. On the other hand, a null (left/right zero) semigroup with an identityadjoined is called a find-unique (find-first/find-last) monoid.
The class of null semigroups is:
It follows that the class of null (left/right zero) semigroups is avariety of universal algebra, and thus avariety of finite semigroups. The variety of finite null semigroups is defined by the identityab =cd.
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