Null semigroup
Encyclopedia
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.
for S is as given below:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a null semigroup (also called a zero semigroup) is a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
with an absorbing element
Absorbing element
In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero 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.
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 tableCayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table...
for S is as given below:
| 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:| 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:| a | b | c | |
|---|---|---|---|
| a | a | b | c |
| b | a | b | c |
| c | a | b | c |