In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative 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 finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups 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 algebraic theory 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 subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.

As in any algebraic theory, one of the main problems of the theory of semigroups is the classification 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 group.

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.

Notations

In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

+Notations Notation Meaning S Arbitrary semigroup E Set of idempotents in S G Group of units in S I Minimal ideal of S V Regular elements of SX Arbitrary set a, b, c Arbitrary elements of S x, y, z Specific elements of S e, f, g Arbitrary elements of E h Specific element of El, m, n Arbitrary positive integersj, k Specific positive integers v, w Arbitrary elements of V 0 Zero element of S 1 Identity element of SS1 S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S a ≤L ba ≤R ba ≤H ba ≤J b S1a ⊆ S1b aS1 ⊆ bS1 S1a ⊆ S1b and aS1 ⊆ bS1 S1aS1 ⊆ S1bS1 L, R, H, D, J Green's relations La, Ra, Ha, Da, Ja Green classes containing a The only power of x which is idempotent. This element exists, assuming the semigroup is (locally) finite. See variety of finite semigroups for more information about this notation. The cardinality of X, assuming X is finite. For example, the definition xab = xba should be read as: There exists x an element of the semigroup such that, for each a and b in the semigroup, xab and xba are equal.

List of special classes of semigroups The third column states whether this set of semigroups forms a variety. And whether the set of finite semigroups of this special class forms a variety of finite semigroups. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.

+List of special classes of semigroups TerminologyDefining property Variety of finite semigroup Reference(s)Finite semigroupS is a finite set.Not infinite FiniteEmpty semigroupS = NoTrivial semigroupCardinality of S is 1.Infinite FiniteMonoid1 ∈ SNoGril p. 3 Band(Idempotent semigroup) a2 = aInfinite FiniteC&P p. 4Rectangular bandA band such that aba = aInfinite FiniteFennemoreNormal bandA band such that abca = acbaInfinite FiniteFennemore Semilattice A commutative band, that is: a2 = a ab = ba Infinite FiniteC&P p. 24 Fennemore Commutative semigroupab = ba Infinite Finite C&P p. 3Archimedean commutative semigroupab = ba There exists x and k such that ak = xb. C&P p. 131Nowhere commutative semigroupab = ba   ⇒   a = bC&P p. 26Left weakly commutativeThere exist x and k such that (ab)k = bx.Nagy p. 59Right weakly commutativeThere exist x and k such that (ab)k = xa.Nagy p. 59Weakly commutative Left and right weakly commutative. That is: There exist x and j such that (ab)j = bx. There exist y and k such that (ab)k = ya.Nagy p. 59Conditionally commutative semigroupIf ab = ba then axb = bxa for all x.Nagy p. 77R-commutative semigroupab R baNagy p. 69–71RC-commutative semigroupR-commutative and conditionally commutativeNagy p. 93–107L-commutative semigroupab L baNagy p. 69–71LC-commutative semigroupL-commutative and conditionally commutativeNagy p. 93–107H-commutative semigroupab H baNagy p. 69–71Quasi-commutative semigroupab = (ba)k for some k.Nagy p. 109Right commutative semigroupxab = xbaNagy p. 137Left commutative semigroupabx = baxNagy p. 137Externally commutative semigroupaxb = bxaNagy p. 175Medial semigroupxaby = xbayNagy p. 119E-k semigroup (k fixed)(ab)k = akbkInfinite FiniteNagy p. 183Exponential semigroup(ab)m = ambm for all mInfinite FiniteNagy p. 183WE-k semigroup (k fixed)There is a positive integer j depending on the couple (a,b) such that (ab)k+j = akbk (ab)j = (ab)jakbkNagy p. 199Weakly exponential semigroupWE-m for all mNagy p. 215Right cancellative semigroupba = ca   ⇒   b = cC&P p. 3Left cancellative semigroupab = ac   ⇒   b = cC&P p. 3Cancellative semigroup Left and right cancellative semigroup, that is ab = ac   ⇒   b = c ba = ca   ⇒   b = cC&P p. 3E-inversive semigroup (E-dense semigroup)There exists x such that ax ∈ E.C&P p. 98Regular semigroupThere exists x such that axa =a.C&P p. 26Regular bandA band such that abaca = abcaInfinite FiniteFennemoreIntra-regular semigroupThere exist x and y such that xa2y = a.C&P p. 121Left regular semigroupThere exists x such that xa2 = a.C&P p. 121Left-regular bandA band such that aba = abInfinite FiniteFennemoreRight regular semigroupThere exists x such that a2x = a.C&P p. 121Right-regular bandA band such that aba = baInfinite FiniteFennemoreCompletely regular semigroupHa is a group.Gril p. 75(inverse) Clifford semigroupA regular semigroup in which all idempotents are central. Equivalently, for finite semigroup: FinitePetrich p. 65k-regular semigroup (k fixed)There exists x such that akxak = ak.HariEventually regular semigroup(π-regular semigroup, Quasi regular semigroup)There exists k and x (depending on a) such that akxak = ak.EdwaShumHigg p. 49Quasi-periodic semigroup, epigroup, group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see Kela for a list)There exists k (depending on a) such that ak belongs to a subgroup of SKelaGril p. 110Higg p. 4Primitive semigroupIf 0 ≠ e and f = ef = fe then e = f.C&P p. 26Unit regular semigroupThere exists u in G such that aua = a.TvmStrongly unit regular semigroupThere exists u in G such that aua = a. e D f ⇒ f = v−1ev for some v in G.TvmOrthodox semigroupThere exists x such that axa = a. E is a subsemigroup of S.Gril p. 57Howi p. 226Inverse semigroupThere exists unique x such that axa = a and xax = x.C&P p. 28Left inverse semigroup (R-unipotent)Ra contains a unique h.Gril p. 382Right inverse semigroup(L-unipotent)La contains a unique h.Gril p. 382Locally inverse semigroup (Pseudoinverse semigroup)There exists x such that axa = a. E is a pseudosemilattice.Gril p. 352M-inversive semigroupThere exist x and y such that baxc = bc and byac = bc.C&P p. 98Abundant semigroupThe classes L*a and R*a, where a L* b if ac = ad ⇔ bc = bd and a R* b if ca = da ⇔ cb = db, contain idempotents.ChenRpp-semigroup(Right principal projective semigroup)The class L*a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.ShumLpp-semigroup(Left principal projective semigroup)The class R*a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.ShumNull semigroup (Zero semigroup)0 ∈ S ab = 0 Equivalently ab = cdInfinite FiniteC&P p. 4Left zero semigroupab = aInfinite FiniteC&P p. 4Left zero bandA left zero semigroup which is a band. That is: ab = a aa = aInfinite FiniteFennemoreLeft group A semigroup which is left simple and right cancellative. The direct product of a left zero semigroup and an abelian group.C&P p. 37, 38Right zero semigroupab = bInfinite FiniteC&P p. 4Right zero bandA right zero semigroup which is a band. That is: ab = b aa = aInfinite FiniteFennemoreRight group A semigroup which is right simple and left cancellative. The direct product of a right zero semigroup and a group.C&P p. 37, 38Right abelian group A right simple and conditionally commutative semigroup.

The direct product of a right zero semigroup and an abelian group.Nagy p. 87Unipotent semigroupE is singleton.Infinite FiniteC&P p. 21Left reductive semigroupIf xa = xb for all x then a = b.C&P p. 9Right reductive semigroupIf ax = bx for all x then a = b.C&P p. 4Reductive semigroupIf xa = xb for all x then a = b. If ax = bx for all x then a = b.C&P p. 4Separative semigroupab = a2 = b2   ⇒   a = bC&P p. 130–131Reversible semigroupSa ∩ Sb ≠ Ø aS ∩ bS ≠ ØC&P p. 34Right reversible semigroupSa ∩ Sb ≠ ØC&P p. 34Left reversible semigroupaS ∩ bS ≠ ØC&P p. 34Aperiodic semigroup There exists k (depending on a) such that ak = ak+1 Equivalently, for finite semigroup: for each a, .KKM p. 29 Pin p. 158ω-semigroupE is countable descending chain under the order a ≤H bGril p. 233–238Left Clifford semigroup(LC-semigroup)aS ⊆ SaShumRight Clifford semigroup(RC-semigroup)Sa ⊆ aSShumOrthogroupHa is a group. E is a subsemigroup of SShumComplete commutative semigroupab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum.Gril p. 110Nilsemigroup (Nilpotent semigroup)0 ∈ S ak = 0 for some integer k which depends on a. Equivalently, for finite semigroup: for each element x and y, .FiniteGril p. 99 Pin p. 148Elementary semigroupab = ba S is of the form G ∪ N where G is a group, and 1 ∈ G N is an ideal, a nilsemigroup, and 0 ∈ NGril p. 111E-unitary semigroupThere exists unique x such that axa = a and xax = x. ea = e   ⇒   a ∈ EGril p. 245Finitely presented semigroupS has a presentation ( X; R ) in which X and R are finite.Gril p. 134Fundamental semigroupEquality on S is the only congruence contained in H.Gril p. 88Idempotent generated semigroupS is equal to the semigroup generated by E.Gril p. 328Locally finite semigroupEvery finitely generated subsemigroup of S is finite.Not infinite FiniteGril p. 161N-semigroupab = ba There exists x and a positive integer n such that a = xbn. ax = ay   ⇒   x = y xa = ya   ⇒   x = y E = ØGril p. 100L-unipotent semigroup (Right inverse semigroup)La contains a unique e.Gril p. 362R-unipotent semigroup (Left inverse semigroup)Ra contains a unique e.Gril p. 362Left simple semigroupLa = SGril p. 57Right simple semigroupRa = SGril p. 57Subelementary semigroupab = ba S = C ∪ N where C is a cancellative semigroup, N is a nilsemigroup or a one-element semigroup. N is ideal of S. Zero of N is 0 of S. For x, y in S and c in C, cx = cy implies that x = y.Gril p. 134Symmetric semigroup(Full transformation semigroup)Set of all mappings of X into itself with composition of mappings as binary operation.C&P p. 2Weakly reductive semigroupIf xz = yz and zx = zy for all z in S then x = y.C&P p. 11Right unambiguous semigroupIf x, y ≥R z then x ≥R y or y ≥R x.Gril p. 170Left unambiguous semigroupIf x, y ≥L z then x ≥L y or y ≥L x.Gril p. 170Unambiguous semigroupIf x, y ≥R z then x ≥R y or y ≥R x. If x, y ≥L z then x ≥L y or y ≥L x.Gril p. 170Left 0-unambiguous0∈ S 0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L yGril p. 178Right 0-unambiguous0∈ S