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Null semigroup

In mathematics a '''null semigroup''' (also called a '''zero semigroup''') is a semigroup with a zero element in which the product of any two elements is zero

Zero element in a semigroup

An element z in a semigroup S is called a zero element of S if for all a in S we have az = za = z Zero element in a semigroup if it exists is unique and it is usually denoted by 0. The zero element of a semigroup S if it exists is also called the zero of S
If a semigroup S has no zero element we can construct a semigroup S0 with zero as follows: Let S0 = S ∪ { 0 } and define a binary operation in S0 as follows:
For all a b in S0 set a . b = ab if a ≠ 0, b ≠ 0 and a . b = 0 otherwise

S0 with this binary operation is a semigroup with zero element the zero element being 0.

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 abc
0 0 0 0 0
'a 0 0 0 0
b 0 0 0 0
c 0 0 0 0

Left zero element and right zero element in a semigroup

An element z in a semigroup S is called a left zero element of S if for all a in S we have za = z Left zero element in a semigroup if it exists need not be unique A left zero element of a semigroup S if it exists is also called a left zero of S
An element z in a semigroup S is called a
right zero element of S if for all a in S we have za = z Right zero element in a semigroup if it exists need not be unique A right zero element of a semigroup S if it exists is also called a right zero of S

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
abc
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
abc
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"