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 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 for all x and y in S we have xy = 0.

Cayley table for a null semigroup

Let S = { 0, a, b, c } be 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 for all x and y in S we have xy = x.

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 for all x and y in S we have xy = y.

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 set of null semigroup is:

  • closed under taking subsemigroup
  • closed under taking quotient of subsemigroup
  • closed under arbitrary direct product.

It follows that the set of null (left/right zero) semigroup 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.


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References
  1. 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.
  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
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