mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (just notation, not necessarily th ...

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. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

, 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'' = be (the underlying set of) a null semigroup. Then the

Cayley table Named after the 19th-century United Kingdom, 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 additi ...

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 is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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

monoid In abstract algebra, 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 . Monoids are semigroups with identity ...

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 In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

of subsemigroup *closed under arbitrary

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

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''.

References

{{reflist Semigroup theory