Null 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.^[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 A. H. Clifford and G. B. 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:

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:

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:

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

• Right group

References

1. A H Clifford; G B Preston (1964). The Algebraic Theory of Semigroups, volume I. mathematical Surveys. Vol. 1 (2 ed.). American Mathematical Society. pp. 3–4. ISBN 978-0-8218-0272-4.
2. 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