In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, an absorbing element (or annihilating 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 ...
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 Clifford and 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'' = be (the underlying set of) 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 In mathematics, an absorbing element (or annihilating 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 ...
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'' = 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 ''xy'' = ''y'' for all ''x'' and ''y'' in ''S''.
Cayley table for a right zero semigroup
Let ''S'' = be a right zero semigroup. Then the Cayley table for ''S'' is as given below:
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.
The class of null semigroups is:
*closed under taking
subsemigroups
*closed under taking
quotient of subsemigroup
*closed under arbitrary
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
s.
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
*
Right group In mathematics, a right group is an algebraic structure consisting of a set together with a binary operation that combines two elements into a third element while obeying the right group axioms. The right group axioms are similar to the group axiom ...
References
{{reflist
Semigroup theory