Null semigroup

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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 the semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.[2]

Null semigroup[edit]

Let S be a semigroup with zero element 0. Then S is called a null semigroup if the following condition is satisfied:

  • For all x, y in S we have xy = 0.

Cayley table for a null semigroup[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

End note[edit]

"In spite of their triviality, these semigroups arise naturally in a number of investigations".[1]

References[edit]

  1. ^ 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. 
  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