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. 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. According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."
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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:
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:
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: