Editing Semigroup (section)

=== Subsemigroups and ideals ===

The semigroup operation induces an operation on the collection of its subsets: given subsets ''A'' and ''B'' of a semigroup ''S'', their product {{math|''A'' · ''B''}}, written commonly as ''AB'', is the set {{math|{{mset| ''ab'' | ''a'' &isin; ''A'' and ''b'' &isin; ''B'' }}.}} (This notion is defined identically as [[Product of group subsets|it is for groups]].) In terms of this operation, a subset ''A'' is called
* a '''subsemigroup''' if ''AA'' is a subset of ''A'',
* a '''right ideal''' if ''AS'' is a subset of ''A'', and
* a '''left ideal''' if ''SA'' is a subset of ''A''.

If ''A'' is both a left ideal and a right ideal then it is called an '''ideal''' (or a '''two-sided ideal''').

If ''S'' is a semigroup, then the intersection of any collection of subsemigroups of ''S'' is also a subsemigroup of ''S''.
So the subsemigroups of ''S'' form a [[complete lattice]].

An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a [[commutative]] semigroup, when it exists, is a group.

[[Green's relations]], a set of five [[equivalence relation]]s that characterise the elements in terms of the [[principal ideal]]s they generate, are important tools for analysing the ideals of a semigroup and related notions of structure.

The subset with the property that every element commutes with any other element of the semigroup is called the '''[[center (algebra)|center]]''' of the semigroup.<ref name="KilpKilʹp2000">{{Cite book|first1=Mati |last1=Kilp |first2=U. |last2=Knauer |first3=Aleksandr V. |last3=Mikhalev |title=Monoids, Acts, and Categories: With Applications to Wreath Products and Graphs : a Handbook for Students and Researchers |url=https://books.google.com/books?id=4gPhmmW-EGcC&pg=PA25 |year=2000 |publisher=Walter de Gruyter |isbn=978-3-11-015248-7 |page=25 |zbl=0945.20036}}</ref> The center of a semigroup is actually a subsemigroup.<ref name="Li͡apin1968">{{Cite book|first=E. S. |last=Li͡apin|title=Semigroups|url=https://books.google.com/books?id=G8pWKPp4tKwC&pg=PA96|year=1968|publisher=American Mathematical Soc.|isbn=978-0-8218-8641-0|page=96}}</ref>