In mathematics, a null semigroup (also called a zero semigroup) is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...

with an

absorbing element 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 ...

, called

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

, 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 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 Named after the 19th century British mathematician Arthur Cayley, a Cayley table 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 multiplicat ...

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 ''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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...

. It follows that the only null (left/right zero)

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...

is the trivial monoid. The class of null semigroups is: *closed under taking subsemigroups *closed under taking

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

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 t ...

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 In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological ...

. The variety of finite null semigroups is defined by the identity ''ab'' = ''cd''.

References

{{reflist Semigroup theory