# Null semigroup

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In [mathematics](/source/mathematics), a '''null semigroup''' (also called a '''zero semigroup''') is a [semigroup](/source/semigroup) with an [absorbing element](/source/absorbing_element), called [zero](/source/Semigroup), 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](/source/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](/source/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](/source/A._H._Clifford) and [G. B. Preston](/source/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](/source/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](/source/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](/source/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](/source/identity_element). It follows that the only null (left/right zero) [monoid](/source/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](/source/Semigroup)
*closed under taking [quotient](/source/Semigroup) of subsemigroup
*closed under arbitrary [direct product](/source/Direct_product)s. 

It follows that the class of null (left/right zero) semigroups is a [variety of universal algebra](/source/variety_(universal_algebra)), and thus a [variety of finite semigroups](/source/variety_of_finite_semigroups). The variety of finite null semigroups is defined by the identity ''ab'' = ''cd''.

==See also==
*[Right group](/source/Right_group)

==References==
{{reflist}}

Category:Semigroup theory

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Adapted from the Wikipedia article [Null semigroup](https://en.wikipedia.org/wiki/Null_semigroup) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Null_semigroup?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.