Special classes of semigroups

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a> is a <a href="/facts/Nonempty_set/ffEk3eA6">nonempty set</a> together with an <a href="/facts/Associative/EQX8lV6r">associative</a> <a href="/facts/Binary_operation/CYtow0bz">binary operation</a>. A special class of semigroups is a <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of <a href="/facts/Semigroup/rgSdk2kt">semigroups</a> satisfying additional <a href="/facts/Property_(philosophy)/mejP0slr">properties</a> or conditions. Thus the class of <a href="/facts/Commutative/WaYxJLxd">commutative</a> semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup.
The class of <a href="/facts/Finite_set/za1newlP">finite</a> semigroups consists of those semigroups for which the <a href="/facts/Underlying_set/TkuXca24">underlying set</a> has finite <a href="/facts/Cardinality/m6VPojwT">cardinality</a>. Members of the class of <a href="/facts/Brandt_semigroup/NcDbeq8H">Brandt semigroups</a> are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.
In the <a href="/facts/Algebra/nMdqczjo">algebraic</a> <a href="/facts/Theory/9vrNS1Rr">theory</a> of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of <a href="/facts/Subset/SJGERmbA">subsets</a> of the <a href="/facts/Underlying_set/TkuXca24">underlying set</a>. The underlying <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> are not assumed to carry any other mathematical <a href="/facts/Structure/xQDPtqDQ">structures</a> like <a href="/facts/Partial_order/qb4a3FAX">order</a> or <a href="/facts/Topology/2EqW47cU">topology</a>.
As in any algebraic theory, one of the main problems of the theory of semigroups is the <a href="/facts/Classification_theorems/NWGgtNm6">classification</a> of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example, the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>.
A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.

Special classes of semigroups open-in-new

Special classes of semigroups