- 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]
Contents
Null semigroup
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
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 End note
"In spite of their triviality, these semigroups arise naturally in a number of investigations".[1]
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 978-0821802724.
- ^ 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 3110152487, p. 19
Categories:- Semigroup theory
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