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.^{[1]} 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.^{[2]} According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."^{[1]}
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:
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:
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:
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
- ^ ^{a} ^{b} A H Clifford; G B Preston (1964). The algebraic theory of semigroups Vol I. mathematical Surveys. 1 (2 ed.). American Mathematical Society. pp. 3–4. ISBN 978-0-8218-0272-4.
- ^ 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