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 a semigroup is a
left zero then the semigroup is called a
left zero semigroup; a
right zero semigroup is defined analogously.
[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, , p. 19]
According to A. H. Clifford and G. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."
Null semigroup
Let
S be a semigroup with zero element 0. Then
S is called a
null semigroup if
xy = 0 for all
x and
y in
S.
Cayley table for a null semigroup
Let
S = {0,
a,
b,
c} be (the underlying set of) a null semigroup. Then the
Cayley table for
S is as given below:
| +Cayley table for a null semigroup | +
!
! 0
! a
! b
! c |
|
|
|
|
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
xy =
x for all
x and
y in
S.
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 |
|
|
|
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
xy =
y for all
x and
y in
S.
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 |
|
|
|
Properties
A non-trivial null (left/right zero) semigroup does not contain an
identity element. It follows that the only null (left/right zero)
monoid is the trivial monoid. On the other hand, a null (left/right zero) semigroup with an identity
adjoined is called a find-unique (find-first/find-last) monoid.
The class of null semigroups is:
-
closed under taking subsemigroups
-
closed under taking quotient of subsemigroup
-
closed under arbitrary direct products.
It follows that the class of null (left/right zero) semigroups is a variety of universal algebra, and thus a variety of finite semigroups. The variety of finite null semigroups is defined by the identity ab = cd.
See also
