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.<ref name="clifford">{{cite book| last=A H Clifford|author2=G B Preston |title=The Algebraic Theory of Semigroups, volume I|publisher=American Mathematical Society| date=1964|edition=2|series=mathematical Surveys|volume=1|pages=3–4|isbn=978-0-8218-0272-4}}</ref> 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.<ref>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, {{isbn|3-11-015248-7}}, p. 19</ref>

According to A. H. Clifford and G. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."<ref name="clifford" />

==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:

{| class="wikitable" style="width: 25%" |+Cayley table for a null semigroup |+ ! ! 0 !''a'' !''b'' !''c'' |- ! 0 | 0 | 0 | 0 | 0 |- ! ''a'' | 0 | 0 | 0 | 0 |- ! ''b'' | 0 | 0 | 0 | 0 |- ! ''c'' | 0 | 0 | 0 | 0 |}

==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:

{| class="wikitable" style="width: 25%" |+Cayley table for a left zero semigroup |+ ! !''a'' !''b'' !''c'' |- ! ''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 ''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:

{| class="wikitable" style="width: 25%" |+Cayley table for a right zero semigroup |+ ! ! ''a'' !''b'' !''c'' |- ! ''a'' | ''a'' | ''b'' | ''c'' |- ! ''b'' | ''a'' | ''b'' | ''c'' |- ! ''c'' | ''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== *Right group

==References== {{reflist}}

Category:Semigroup theory