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 variety 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
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
0	Zero element of S
1	Identity element of S
S^1	S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S
L, R, H, D, J	Green's relations
La, Ra, Ha, Da, Ja	Green classes containing a
a ≤L b a ≤R b a ≤H b	S^1a ⊆ S^1b aS^1 ⊆ bS^1 S^1a ⊆ S^1b and aS^1 ⊆ bS^1

List of special classes of semigroups

List of special classes of semigroups

Terminology	Defining property	Reference(s)
Finite semigroup	S is a finite set.
Empty semigroup	S = $\emptyset$
Trivial semigroup	Cardinality of S is 1.
Monoid	1 ∈ S	Gril p. 3
Band (Idempotent semigroup)	a^2 = a	C&P p. 4
Semilattice	a^2 = a ab = ba	C&P p. 24
Commutative semigroup	ab = ba	C&P p. 3
Archimedean commutative semigroup	ab = ba There exists x and k such that a^k = 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	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 = a^kb^k	Nagy p. 183
Exponential semigroup	(ab)^m = a^mb^m for all m	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 = a^kb^k (ab)^j = (ab)^ja^kb^k	Nagy p. 199
Weakly exponential semigroup	WE-m for all m	Nagy p. 215
Cancellative semigroup	ax = ay   ⇒   x = y xa = ya   ⇒   x = y	C&P p. 3
Right cancellative semigroup	xa = ya   ⇒   x = y	C&P p. 3
Left cancellative semigroup	ax = ay   ⇒   x = y	C&P p. 3
E-inversive semigroup	There exists x such that ax ∈ E.	C&P p. 98
Regular semigroup	There exists x such that axa =a.	C&P p. 26
Intra-regular semigroup	There exist x and y such that xa^2y = a.	C&P p. 121
Left regular semigroup	There exists x such that xa^2 = a.	C&P p. 121
Right regular semigroup	There exists x such that a^2x = a.	C&P p. 121
Completely regular semigroup	Ha is a group.	Gril p. 75
(inverse) Clifford semigroup	A regular semigroup in which all idempotents are central.	Petrich p. 65
k-regular semigroup (k fixed)	There exists x such that a^kxa^k = a^k.	Hari
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup)	There exists k and x (depending on a) such that a^kxa^k = a^k.	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 a^k 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
Pseudoinverse semigroup (Locally inverse semigroup)	There exists x such that axa = a. E is a pseudosemilattice.	Gril p. 352
Abundant semigroups	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	C&P p. 4
Zero semigroup (Null semigroup)	0 ∈ S ab = 0	C&P p. 4
Left zero semigroup	ab = a	C&P p. 4
Right zero semigroup	ab = b	C&P p. 4
Unipotent semigroup	E is singleton.	C&P p. 21
Left reductive semigroup	If xa = xb for all x implies a = b.	C&P p. 9
Right reductive semigroup	If ax = bx for all x implies a = b.	C&P p. 4
Reductive semigroup	If xa = xb for all x implies a = b. If ax = bx for all x implies a = b.	C&P p. 4
Separative semigroup	ab = a^2 = b^2   ⇒   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 a^k = a^k+1	KKM p. 29
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
LC-semigroup (Left Clifford semigroup)	aS ⊆ Sa	Shum
RC-semigroup (Right Clifford semigroup)	Sa ⊆ aS	Shum
Orthogroup	Ha is a group. E is a subsemigroup of S	Shum
Complete commutative semigroup	ab = ba a^k is in a subgroup of S for some k. Every nonempty subset of E has an infimum.	Gril p. 110
Nilsemigroup	0 ∈ S a^k = 0 for some k.	Gril p. 99
Elementary semigroup	ab = ba S = G ∪ N where G is a group, N is a nilsemigroup or a one-element semigroup. N is ideal of S. Identity of G is 1 of S and zero of N is 0 of S.	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.	Gril p. 161
N-semigroup	ab = ba There exists x and a positive integer n such that a = xb^n. 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 ∈ bS^1   ⇒   a^n ∈ b^2S^1 for some n.	Nagy p. 35
Right Putcha semigroup	a ∈ S^1b   ⇒   a^n ∈ S^1b^2 for some n.	Nagy p. 35
Putcha semigroup	a ∈ S^1b S^1   ⇒   a^n ∈ S^1b^2S^1 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 S^2 ≠ 0 If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0. 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) = S^1aS^1, I(a) = J(a) − Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.	C&P p. 71–75
Simple semigroup	Ja = S. (There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.)	C&P p. 5 Higg p. 16
0-simple semigroup	0 ∈ S S^2 ≠ 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 S^2 ≠ 0 If A ⊆ S is such that SA ⊆ A then A = 0.	C&P p. 67
Right 0-simple semigroup	0 ∈ S S^2 ≠ 0 If A ⊆ S is such that AS ⊆ A then A = 0.	C&P p. 67
Cyclic semigroup (Monogenic semigroup)	S = { w, w^2, w^3, ... } for some w in S	C&P p. 19
Monogenic semigroup (Cyclic semigroup)	S = { w, w^2, w^3, ... } for some w in S	C&P p. 19
Periodic semigroup	{ a, a^2, a^3, ... } is a finite set.	C&P p. 20
Bicyclic semigroup	1 ∈ S S generated by { x1, x2 } with x1x2 = 1.	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 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	G^0 a group G with 0 adjoined. P : Λ × I → G^0 a map. Define an operation in I × G^0 × Λ by ( i, g, λ ) ( j, h, μ ) = ( i, g P( λ, j ) h, μ ). ( I, G^0, Λ )/( I × { 0 } × Λ ) is the Rees matrix semigroup M^0 ( G^0; 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
*-semigroup (Semigroup with involution)	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.

References

[C&P]	A H Clifford, G B Preston (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. ISBN 978-0-8218-0272-4
[C&P II]	A H Clifford, G B Preston (1967). The Algebraic Theory of Semigroups Vol. II (Second Edition). American Mathematical Society. ISBN 0-8218-0272-0
[Chen]	Hui Chen (2006), "Construction of a kind of abundant semigroups", Mathematical Communications (11), 165–171 (Accessed on 25 April 2009)
[Delg]	M Delgado, et al., Numerical semigroups, (Accessed on 27 April 2009)
[Edwa]	P M Edwards (1983), "Eventually regular semigroups", Bulletin of Australian Mathematical Society 28, 23–38
[Gril]	P A Grillet (1995). Semigroups. CRC Press. ISBN 978-0-8247-9662-4
[Hari]	K S Harinath (1979), "Some results on k-regular semigroups", Indian Journal of Pure and Applied Mathematics 10(11), 1422–1431
[Howi]	J M Howie (1995), Fundamentals of Semigroup Theory, Oxford University Press
[Nagy]	Attila Nagy (2001). Special Classes of Semigroups. Springer. ISBN 978-0-7923-6890-8
[Pet]	M Petrich, N R Reilly (1999). Completely regular semigroups. John Wiley & Sons. ISBN 978-0-471-19571-9
[Shum]	K P Shum "Rpp semigroups, its generalizations and special subclasses" in Advances in Algebra and Combinatorics edited by K P Shum et al. (2008), World Scientific, ISBN 981-279-000-4 (pp. 303–334)
[Tvm]	Proceedings of the International Symposium on Theory of Regular Semigroups and Applications, University of Kerala, Thiruvananthapuram, India, 1986
[Kela]	A. V. Kelarev, Applications of epigroups to graded ring theory, Semigroup Forum, Volume 50, Number 1 (1995), 327-350 doi:10.1007/BF02573530
[KKM]	Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, Expositions in Mathematics 29, Walter de Gruyter, Berlin, ISBN 978-3-11-015248-7.
[Higg]	Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. ISBN 978-0-19-853577-5.