Special classes of semigroups

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 S
X	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 E
l, m, n	Arbitrary positive integers
j, k	Specific positive integers
v, w	Arbitrary elements of V
0	Zero element of S
1	Identity element of S
S1	S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S
a ≤L b a ≤R b a ≤H b a ≤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
$x^{\omega }$	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.
$|X|$	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

Terminology	Defining property	Variety of finite semigroup	Reference(s)
Finite semigroup	S is a finite set.	Not infinite Finite
Empty semigroup	S = $\emptyset$	No
Trivial semigroup	Cardinality of S is 1.	Infinite Finite
Monoid	1 ∈ S	No	Gril p. 3
Band (Idempotent semigroup)	a2 = a	Infinite Finite	C&P p. 4
Rectangular band	A band such that abca = acba	Infinite Finite	Fennemore
Semilattice	A commutative band, that is: a2 = a ab = ba	Infinite Finite	C&P p. 24 Fennemore
Commutative semigroup	ab = ba	Infinite Finite	C&P p. 3
Archimedean commutative semigroup	ab = ba There exists x and k such that ak = xb.		C&P p. 131
Nowhere commutative semigroup	ab = ba   ⇒   a = b		C&P p. 26
Left weakly commutative	There exist x and k such that (ab)k = bx.		Nagy p. 59
Right weakly commutative	There exist x and k such that (ab)k = xa.		Nagy p. 59
Weakly 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. 59
Conditionally commutative semigroup	If ab = ba then axb = bxa for all x.		Nagy p. 77
R-commutative semigroup	ab R ba		Nagy p. 69–71
RC-commutative semigroup	R-commutative and conditionally commutative		Nagy p. 93–107
L-commutative semigroup	ab L ba		Nagy p. 69–71
LC-commutative semigroup	L-commutative and conditionally commutative		Nagy p. 93–107
H-commutative semigroup	ab H ba		Nagy p. 69–71
Quasi-commutative semigroup	ab = (ba)k for some k.		Nagy p. 109
Right commutative semigroup	xab = xba		Nagy p. 137
Left commutative semigroup	abx = bax		Nagy p. 137
Externally commutative semigroup	axb = bxa		Nagy p. 175
Medial semigroup	xaby = xbay		Nagy p. 119
E-k semigroup (k fixed)	(ab)k = akbk	Infinite Finite	Nagy p. 183
Exponential semigroup	(ab)m = ambm for all m	Infinite Finite	Nagy p. 183
WE-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)jakbk		Nagy p. 199
Weakly exponential semigroup	WE-m for all m		Nagy p. 215
Right cancellative semigroup	ba = ca   ⇒   b = c		C&P p. 3
Left cancellative semigroup	ab = ac   ⇒   b = c		C&P p. 3
Cancellative semigroup	Left and right cancellative semigroup, that is ab = ac   ⇒   b = c ba = ca   ⇒   b = c		C&P p. 3
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Regular semigroup	There exists x such that axa =a.		C&P p. 26
Regular band	A band such that abaca = abca	Infinite Finite	Fennemore
Intra-regular semigroup	There exist x and y such that xa2y = a.		C&P p. 121
Left regular semigroup	There exists x such that xa2 = a.		C&P p. 121
Left-regular band	A band such that aba = ab	Infinite Finite	Fennemore
Right regular semigroup	There exists x such that a2x = a.		C&P p. 121
Right-regular band	A band such that aba = ba	Infinite Finite	Fennemore
Completely regular semigroup	Ha is a group.		Gril p. 75
(inverse) Clifford semigroup	A regular semigroup in which all idempotents are central. Equivalently, for finite semigroup: $a^{\omega }b=ba^{\omega }$	Finite	Petrich p. 65
k-regular semigroup (k fixed)	There exists x such that akxak = ak.		Hari
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup)	There exists k and x (depending on a) such that akxak = ak.		Edwa Shum Higg p. 49
Quasi-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 S		Kela Gril p. 110 Higg p. 4
Primitive semigroup	If 0 ≠ e and f = ef = fe then e = f.		C&P p. 26
Unit regular semigroup	There exists u in G such that aua = a.		Tvm
Strongly unit regular semigroup	There exists u in G such that aua = a. e D f ⇒ f = v−1ev for some v in G.		Tvm
Orthodox semigroup	There exists x such that axa = a. E is a subsemigroup of S.		Gril p. 57 Howi p. 226
Inverse semigroup	There exists unique x such that axa = a and xax = x.		C&P p. 28
Left inverse semigroup (R-unipotent)	Ra contains a unique h.		Gril p. 382
Right inverse semigroup (L-unipotent)	La contains a unique h.		Gril p. 382
Locally inverse semigroup (Pseudoinverse semigroup)	There exists x such that axa = a. E is a pseudosemilattice.		Gril p. 352
M-inversive semigroup	There exist x and y such that baxc = bc and byac = bc.		C&P p. 98
Abundant semigroup	The 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.		Chen
Rpp-semigroup (Right principal projective semigroup)	The class L*a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.		Shum
Lpp-semigroup (Left principal projective semigroup)	The class R*a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.		Shum
Null semigroup (Zero semigroup)	0 ∈ S ab = 0 Equivalently ab = cd	Infinite Finite	C&P p. 4
Left zero semigroup	ab = a	Infinite Finite	C&P p. 4
Left zero band	A left zero semigroup which is a band. That is: ab = a aa = a	Infinite Finite	Fennemore
Left 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, 38
Right zero semigroup	ab = b	Infinite Finite	C&P p. 4
Right zero band	A right zero semigroup which is a band. That is: ab = b aa = a	Infinite Finite	Fennemore
Right 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, 38
Right abelian group	A right simple and conditionally commutative semigroup. The direct product of a right zero semigroup and an abelian group.		Nagy p. 87
Unipotent semigroup	E is singleton.	Infinite Finite	C&P p. 21
Left reductive semigroup	If xa = xb for all x then a = b.		C&P p. 9
Right reductive semigroup	If ax = bx for all x then a = b.		C&P p. 4
Reductive semigroup	If xa = xb for all x then a = b. If ax = bx for all x then a = b.		C&P p. 4
Separative semigroup	ab = a2 = b2   ⇒   a = b		C&P p. 130–131
Reversible semigroup	Sa ∩ Sb ≠ Ø aS ∩ bS ≠ Ø		C&P p. 34
Right reversible semigroup	Sa ∩ Sb ≠ Ø		C&P p. 34
Left reversible semigroup	aS ∩ bS ≠ Ø		C&P p. 34
Aperiodic semigroup	There exists k (depending on a) such that ak = ak+1 Equivalently, for finite semigroup: for each a, $a^{\omega }a=a^{\omega }$.		KKM p. 29 Pin p. 158
ω-semigroup	E is countable descending chain under the order a ≤H b		Gril p. 233–238
Left Clifford semigroup (LC-semigroup)	aS ⊆ Sa		Shum
Right Clifford semigroup (RC-semigroup)	Sa ⊆ aS		Shum
Orthogroup	Ha is a group. E is a subsemigroup of S		Shum
Complete commutative semigroup	ab = ba ak is in a subgroup of S for some k. Every nonempty subset of E has an infimum.		Gril p. 110
Nilsemigroup (Nilpotent semigroup)	0 ∈ S ak = 0 for some integer k which depends on a. Equivalently, for finite semigroup: for each element x and y, $yx^{\omega }=x^{\omega }=x^{\omega }y$.	Finite	Gril p. 99 Pin p. 148
Elementary semigroup	ab = ba S is of the form G ∪ N where G is a group, and 1 ∈ G N is an ideal, a nilsemigroup, and 0 ∈ N		Gril p. 111
E-unitary semigroup	There exists unique x such that axa = a and xax = x. ea = e   ⇒   a ∈ E		Gril p. 245
Finitely presented semigroup	S has a presentation ( X; R ) in which X and R are finite.		Gril p. 134
Fundamental semigroup	Equality on S is the only congruence contained in H.		Gril p. 88
Idempotent generated semigroup	S is equal to the semigroup generated by E.		Gril p. 328
Locally finite semigroup	Every finitely generated subsemigroup of S is finite.	Not infinite Finite	Gril p. 161
N-semigroup	ab = ba There exists x and a positive integer n such that a = xbn. ax = ay   ⇒   x = y xa = ya   ⇒   x = y E = Ø		Gril p. 100
L-unipotent semigroup (Right inverse semigroup)	La contains a unique e.		Gril p. 362
R-unipotent semigroup (Left inverse semigroup)	Ra contains a unique e.		Gril p. 362
Left simple semigroup	La = S		Gril p. 57
Right simple semigroup	Ra = S		Gril p. 57
Subelementary semigroup	ab = 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. 134
Symmetric semigroup (Full transformation semigroup)	Set of all mappings of X into itself with composition of mappings as binary operation.		C&P p. 2
Weakly reductive semigroup	If xz = yz and zx = zy for all z in S then x = y.		C&P p. 11
Right unambiguous semigroup	If x, y ≥R z then x ≥R y or y ≥R x.		Gril p. 170
Left unambiguous semigroup	If x, y ≥L z then x ≥L y or y ≥L x.		Gril p. 170
Unambiguous semigroup	If 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. 170
Left 0-unambiguous	0∈ S 0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y		Gril p. 178
Right 0-unambiguous	0∈ S 0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y		Gril p. 178
0-unambiguous semigroup	0∈ S 0 ≠ x ≤L y, z   ⇒   y ≤L z or z ≤L y 0 ≠ x ≤R y, z   ⇒   y ≤L z or z ≤R y		Gril p. 178
Left Putcha semigroup	a ∈ bS1   ⇒   an ∈ b2S1 for some n.		Nagy p. 35
Right Putcha semigroup	a ∈ S1b   ⇒   an ∈ S1b2 for some n.		Nagy p. 35
Putcha semigroup	a ∈ S1b S1   ⇒   an ∈ S1b2S1 for some positive integer n		Nagy p. 35
Bisimple semigroup (D-simple semigroup)	Da = S		C&P p. 49
0-bisimple semigroup	0 ∈ S S - {0} is a D-class of S.		C&P p. 76
Completely simple semigroup	There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A. There exists h in E such that whenever hf = f and fh = f we have h = f.		C&P p. 76
Completely 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0 or A = S. There exists non-zero h in E such that whenever hf = f, fh = f and f ≠ 0 we have h = f.		C&P p. 76
D-simple semigroup (Bisimple semigroup)	Da = S		C&P p. 49
Semisimple semigroup	Let J(a) = S1aS1, I(a) = J(a) − Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.		C&P p. 71–75
$\mathbf {CS}$: Simple semigroup	Ja = S. (There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.), equivalently, for finite semigroup: $a^{\omega }a=a$ and $(aba)^{\omega }=a^{\omega }$.	Finite	C&P p. 5 Higg p. 16 Pin pp. 151, 158
0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0.		C&P p. 67
Left 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that SA ⊆ A then A = 0.		C&P p. 67
Right 0-simple semigroup	0 ∈ S S2 ≠ 0 If A ⊆ S is such that AS ⊆ A then A = 0.		C&P p. 67
Cyclic semigroup (Monogenic semigroup)	S = { w, w2, w3, ... } for some w in S	Not infinite Not finite	C&P p. 19
Periodic semigroup	{ a, a2, a3, ... } is a finite set.	Not infinite Finite	C&P p. 20
Bicyclic semigroup	1 ∈ S S admits the presentation $\langle x,y\mid xy=1\rangle$.		C&P p. 43–46
Full transformation semigroup TX (Symmetric semigroup)	Set of all mappings of X into itself with composition of mappings as binary operation.		C&P p. 2
Rectangular band	A band such that aba = a Equivalently abc = ac	Infinite Finite	Fennemore
Rectangular semigroup	Whenever three of ax, ay, bx, by are equal, all four are equal.		C&P p. 97
Symmetric inverse semigroup IX	The semigroup of one-to-one partial transformations of X.		C&P p. 29
Brandt semigroup	0 ∈ S ( ac = bc ≠ 0 or ca = cb ≠ 0 )   ⇒   a = b ( ab ≠ 0 and bc ≠ 0 )   ⇒   abc ≠ 0 If a ≠ 0 there exist unique x, y, z, such that xa = a, ay = a, za = y. ( e ≠ 0 and f ≠ 0 )   ⇒   eSf ≠ 0.		C&P p. 101
Free semigroup FX	Set of finite sequences of elements of X with the operation ( x1, ..., xm ) ( y1, ..., yn ) = ( x1, ..., xm, y1, ..., yn )		Gril p. 18
Rees matrix semigroup	G0 a group G with 0 adjoined. P : Λ × I → G0 a map. Define an operation in I × G0 × Λ by ( i, g, λ ) ( j, h, μ ) = ( i, g P( λ, j ) h, μ ). ( I, G0, Λ )/( I × { 0 } × Λ ) is the Rees matrix semigroup M0 ( G0; I, Λ ; P ).		C&P p.88
Semigroup of linear transformations	Semigroup of linear transformations of a vector space V over a field F under composition of functions.		C&P p.57
Semigroup of binary relations BX	Set of all binary relations on X under composition		C&P p.13
Numerical semigroup	0 ∈ S ⊆ N = { 0,1,2, ... } under + . N - S is finite		Delg
Semigroup with involution (*-semigroup)	There exists a unary operation a → a* in S such that a** = a and (ab)* = b*a*.		Howi
Baer–Levi semigroup	Semigroup of one-to-one transformations f of X such that X − f ( X ) is infinite.		C&P II Ch.8
U-semigroup	There exists a unary operation a → a’ in S such that ( a’)’ = a.		Howi p.102
I-semigroup	There exists a unary operation a → a’ in S such that ( a’)’ = a and aa’a = a.		Howi p.102
Semiband	A regular semigroup generated by its idempotents.		Howi p.230
Group	There exists h such that for all a, ah = ha = a. There exists x (depending on a) such that ax = xa = h.	Not infinite Finite
Topological semigroup	A semigroup which is also a topological space. Such that the semigroup product is continuous.	Not applicable	Pin p. 130
Syntactic semigroup	The smallest finite monoid which can recognize a subset of another semigroup.		Pin p. 14
$\mathbf {R}$: the R-trivial monoids	R-trivial. That is, each R-equivalence class is trivial. Equivalently, for finite semigroup: $(ab)^{\omega }a=(ab)^{\omega }$.	Finite	Pin p. 158
$\mathbf {L}$: the L-trivial monoids	L-trivial. That is, each L-equivalence class is trivial. Equivalently, for finite monoids, $b(ab)^{\omega }=(ab)^{\omega }$.	Finite	Pin p. 158
$\mathbf {J}$: the J-trivial monoids	Monoids which are J-trivial. That is, each J-equivalence class is trivial. Equivalently, the monoids which are L-trivial and R-trivial.	Finite	Pin p. 158
$\mathbf {R_{1}}$: idempotent and R-trivial monoids	R-trivial. That is, each R-equivalence class is trivial. Equivalently, for finite monoids: aba = ab.	Finite	Pin p. 158
$\mathbf {L_{1}}$: idempotent and L-trivial monoids	L-trivial. That is, each L-equivalence class is trivial. Equivalently, for finite monoids: aba = ba.	Finite	Pin p. 158
$\mathbb {D} \mathbf {S}$: Semigroup whose regular D are semigroup	Equivalently, for finite monoids: $(a^{\omega }a^{\omega }a^{\omega })^{\omega }=a^{\omega }$. Equivalently, regular H-classes are groups, Equivalently, v≤Ja implies v R va and v L av Equivalently, for each idempotent e, the set of a such that e≤Ja is closed under product (i.e. this set is a subsemigroup) Equivalently, there exists no idempotent e and f such that e J f but not ef J e Equivalently, the monoid $B_{2}^{1}$ does not divide $S\times S$	Finite	Pin pp. 154, 155, 158
$\mathbb {D} \mathbf {A}$: Semigroup whose regular D are aperiodic semigroup	Each regular D-class is an aperiodic semigroup Equivalently, every regular D-class is a rectangular band Equivalently, regular D-class are semigroup, and furthermore S is aperiodic Equivalently, for finite monoid: regular D-class are semigroup, and furthermore $aa^{\omega }=a^{\omega }$ Equivalently, e≤Ja implies eae = e Equivalently, e≤Jf implies efe = e.	Finite	Pin p. 156, 158
$\ell \mathbf {1}$/$\mathbf {K}$: Lefty trivial semigroup	e: eS = e, Equivalently, I is a left zero semigroup equal to E, Equivalently, for finite semigroup: I is a left zero semigroup equals $S^{|S|}$, Equivalently, for finite semigroup: $a_{1}\dots a_{n}y=a_{1}\dots a_{n}$, Equivalently, for finite semigroup: $a^{\omega }b=a^{\omega }$.	Finite	Pin pp. 149, 158
$\mathbf {r1}$/$\mathbf {D}$: Right trivial semigroup	e: Se = e, Equivalently, I is a right zero semigroup equal to E, Equivalently, for finite semigroup: I is a right zero semigroup equals $S^{|S|}$, Equivalently, for finite semigroup: $ba_{1}\dots a_{n}=a_{1}\dots a_{n}$, Equivalently, for finite semigroup: $ba^{\omega }=a^{\omega }$.	Finite	Pin pp. 149, 158
$\mathbb {L} \mathbf {1}$: Locally trivial semigroup	eSe = e, Equivalently, I is equal to E, Equivalently, eaf = ef, Equivalently, for finite semigroup: $ya_{1}\dots a_{n}=a_{1}\dots a_{n}$, Equivalently, for finite semigroup: $a_{1}\dots a_{n}ya_{1}\dots a_{n}=a_{1}\dots a_{n}$, Equivalently, for finite semigroup: $a^{\omega }ba^{\omega }=a^{\omega }$.	Finite	Pin pp. 150, 158
$\mathbb {L} \mathbf {G}$: Locally groups	eSe is a group, Equivalently, E⊆I, Equivalently, for finite semigroup: $(a^{\omega }ba^{\omega })^{\omega }=a^{\omega }$.	Finite	Pin pp. 151, 158

List of special classes of ordered semigroups

Terminology	Defining property	Variety	Reference(s)
Ordered semigroup	A semigroup with a partial order relation ≤, such that a ≤ b implies c•a ≤ c•b and a•c ≤ b•c	Finite	Pin p. 14
$\mathbf {N} ^{+}$	Nilpotent finite semigroups, with $a\leq b^{\omega }$	Finite	Pin pp. 157, 158
$\mathbf {N} ^{-}$	Nilpotent finite semigroups, with $b^{\omega }\leq a$	Finite	Pin pp. 157, 158
$\mathbf {J} _{1}^{+}$	Semilattices with $1\leq a$	Finite	Pin pp. 157, 158
$\mathbf {J} _{1}^{-}$	Semilattices with $a\leq 1$	Finite	Pin pp. 157, 158
$\mathbb {L} \mathbf {J} _{1}^{+}$ locally positive J-trivial semigroup	Finite semigroups satisfying $a^{\omega }\leq a^{\omega }ba^{\omega }$	Finite	Pin pp. 157, 158

References

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[Tvm]		Proceedings of the International Symposium on Theory of Regular Semigroups and Applications, University of Kerala, Thiruvananthapuram, India, 1986
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