Null semigroup explained

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."

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 = be (the underlying set of) a null semigroup. Then the Cayley table for S is as given below:

+! ! 0!a!b!c
00000
a0000
b0000
c0000

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 = be a left zero semigroup. Then the Cayley table for S is as given below:

+! !a!b!c
aaaa
bbbb
cccc

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 = be a right zero semigroup. Then the Cayley table for S is as given below:

+! ! a!b!c
aabc
babc
cabc

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:

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

Notes and References

  1. Book: A H Clifford. G B Preston . The Algebraic Theory of Semigroups, volume I. American Mathematical Society. 1964. 2. mathematical Surveys. 1. 3–4. 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,, p. 19