special classes of semigroups

|+Notations

| Arbitrary semigroup

| Set of idempotents in S

| Group of units in S

| Minimal ideal of S

| Regular elements of S

| Arbitrary set

| Arbitrary elements of S

| Specific elements of S

| Arbitrary elements of E

| Specific element of E

| Arbitrary positive integers

| Specific positive integers

| Arbitrary elements of V

| Zero element of S

| Identity element of S

| S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S

| S¹a ⊆ S¹b
aS¹ ⊆ bS¹
S¹a ⊆ S¹b and aS¹ ⊆ bS¹
S¹aS¹ ⊆ S¹bS¹

| Green's relations

| 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.

|+List of special classes of semigroups

|

• S is a finite set.

|

• Not infinite
• Finite

|

|

• S = $\backslash emptyset$

|No

|

|

• Cardinality of S is 1.

|

• Infinite
• Finite

|

|

• 1 ∈ S

|No

|Gril p. 3

|

• a² = a

|

• Infinite
• Finite

|C&P p. 4

|

• A band such that abca = acba

|

• Infinite
• Finite

|Fennemore

| A commutative band, that is:

• a² = a
• ab = ba

|

• Infinite
• Finite

|

• C&P p. 24
• Fennemore

|

• ab = ba

|

• Infinite
• Finite

| C&P p. 3

|

• ab = ba
• There exists x and k such that a^k = xb.

|

|C&P p. 131

|

• ab = ba   ⇒   a = b

|

|C&P p. 26

|

• There exist x and k such that (ab)^k = bx.

|

|Nagy p. 59

|

• There exist x and k such that (ab)^k = xa.

|

|Nagy p. 59

| 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

|

• If ab = ba then axb = bxa for all x.

|

|Nagy p. 77

|

• ab R ba

|

|Nagy p. 69–71

|

• R-commutative and conditionally commutative

|

|Nagy p. 93–107

|

• ab L ba

|

|Nagy p. 69–71

|

• L-commutative and conditionally commutative

|

|Nagy p. 93–107

|

• ab H ba

|

|Nagy p. 69–71

|

• ab = (ba)^k for some k.

|

|Nagy p. 109

|

• xab = xba

|

|Nagy p. 137

|

• abx = bax

|

|Nagy p. 137

|

• axb = bxa

|

|Nagy p. 175

|

• xaby = xbay

|

|Nagy p. 119

|

• (ab)^k = a^kb^k

|

• Infinite
• Finite

|Nagy p. 183

|

• (ab)^m = a^mb^m for all m

|

• Infinite
• Finite

|Nagy p. 183

|

• 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

|

• WE-m for all m

|

|Nagy p. 215

|

• ba = ca   ⇒   b = c

|

|C&P p. 3

|

• ab = ac   ⇒   b = c

|

|C&P p. 3

| Left and right cancellative semigroup, that is

• ab = ac   ⇒   b = c
• ba = ca   ⇒   b = c

|

|C&P p. 3

|

• There exists x such that ax ∈ E.

|

|C&P p. 98

|

• There exists x such that axa =a.

|

|C&P p. 26

|

• A band such that abaca = abca

|

• Infinite
• Finite

|Fennemore

|

• There exist x and y such that xa²y = a.

|

|C&P p. 121

|

• There exists x such that xa² = a.

|

|C&P p. 121

|

• A band such that aba = ab

|

• Infinite
• Finite

|Fennemore

|

• There exists x such that a²x = a.

|

|C&P p. 121

|

• A band such that aba = ba

|

• Infinite
• Finite

|Fennemore

|

• H_a is a group.

|

|Gril p. 75

|

• A regular semigroup in which all idempotents are central.
• Equivalently, for finite semigroup: $a^\backslash omega\; b=ba^\backslash omega$

|

• Finite

|Petrich p. 65

|

• There exists x such that a^kxa^k = a^k.

|

|Hari

|

• There exists k and x (depending on a) such that a^kxa^k = a^k.

|

|Edwa
Shum
Higg p. 49

|

• There exists k (depending on a) such that a^k belongs to a subgroup of S

|

|Kela
Gril p. 110
Higg p. 4

|

• If 0 ≠ e and f = ef = fe then e = f.

|

|C&P p. 26

|

• There exists u in G such that aua = a.

|

|Tvm

|

• There exists u in G such that aua = a.
• e D f ⇒ f = v⁻¹ev for some v in G.

|

|Tvm

|

• There exists x such that axa = a.
• E is a subsemigroup of S.

|

|Gril p. 57
Howi p. 226

|

• There exists unique x such that axa = a and xax = x.

|

|C&P p. 28

|

• R_a contains a unique h.

|

|Gril p. 382

|

• L_a contains a unique h.

|

|Gril p. 382

|

• There exists x such that axa = a.
• E is a pseudosemilattice.

|

|Gril p. 352

|

• There exist x and y such that baxc = bc and byac = bc.

|

|C&P p. 98

|

• 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

|

• The class L*_a, where a L* b if ac = ad ⇔ bc = bd, contains at least one idempotent.

|

|Shum

|

• The class R*_a, where a R* b if ca = da ⇔ cb = db, contains at least one idempotent.

|

|Shum

|

• 0 ∈ S
• ab = 0
• Equivalently ab = cd

|

• Infinite
• Finite

|C&P p. 4

|

• ab = a

|

• Infinite
• Finite

|C&P p. 4

|A left zero semigroup which is a band. That is:

• ab = a
• aa = a

|

• Infinite
• Finite

|

• Fennemore

|

• 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

|

• ab = b

|

• Infinite
• Finite

|C&P p. 4

|A right zero semigroup which is a band. That is:

• ab = b
• aa = a

|

• Infinite
• Finite

|Fennemore

|

• 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

|

• A right simple and conditionally commutative semigroup.

• The direct product of a right zero semigroup and an abelian group.

|

|Nagy p. 87

|

• E is singleton.

|

• Infinite
• Finite

|C&P p. 21

|

• If xa = xb for all x then a = b.

|

|C&P p. 9

|

• If ax = bx for all x then a = b.

|

|C&P p. 4

|

• If xa = xb for all x then a = b.
• If ax = bx for all x then a = b.

|

|C&P p. 4

|

• ab = a² = b²   ⇒   a = b

|

|C&P p. 130–131

|

• Sa ∩ Sb ≠ Ø
• aS ∩ bS ≠ Ø

|

|C&P p. 34

|

• Sa ∩ Sb ≠ Ø

|

|C&P p. 34

|

• aS ∩ bS ≠ Ø

|

|C&P p. 34

|

• There exists k (depending on a) such that a^k = a^k+1
• Equivalently, for finite semigroup: for each a, $a^\backslash omega\; a=a^\backslash omega$.

|

|

• KKM p. 29
• Pin p. 158

|

• E is countable descending chain under the order a ≤_H b

|

|Gril p. 233–238

|

• aS ⊆ Sa

|

|Shum

|

• Sa ⊆ aS

|

|Shum

|

• H_a is a group.
• E is a subsemigroup of S

|

|Shum

|

• ab = ba
• a^k is in a subgroup of S for some k.
• Every nonempty subset of E has an infimum.

|

|Gril p. 110

|

• 0 ∈ S
• a^k = 0 for some integer k which depends on a.
• Equivalently, for finite semigroup: for each element x and y, $yx^\backslash omega\; =x^\backslash omega=x^\backslash omega\; y$.

|

• Finite

|

• Gril p. 99
• Pin p. 148

|

• 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

|

• There exists unique x such that axa = a and xax = x.
• ea = e   ⇒   a ∈ E

|

|Gril p. 245

|

• S has a presentation ( X; R ) in which X and R are finite.

|

|Gril p. 134

|

• Equality on S is the only congruence contained in H.

|

|Gril p. 88

|

• S is equal to the semigroup generated by E.

|

|Gril p. 328

|

• Every finitely generated subsemigroup of S is finite.

|

• Not infinite
• Finite

|Gril p. 161

|

• 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_a contains a unique e.

|

|Gril p. 362

|

• R_a contains a unique e.

|

|Gril p. 362

|

• L_a = S

|

|Gril p. 57

|

• R_a = S

|

|Gril p. 57

|

• 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

|

• Set of all mappings of X into itself with composition of mappings as binary operation.

|

|C&P p. 2

|

• If xz = yz and zx = zy for all z in S then x = y.

|

|C&P p. 11

|

• If x, y ≥_R z then x ≥_R y or y ≥_R x.

|

|Gril p. 170

|

• If x, y ≥_L z then x ≥_L y or y ≥_L x.

|

|Gril p. 170

|

• 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

|

• 0∈ S
• 0 ≠ x ≤_L y, z   ⇒   y ≤_L z or z ≤_L y

|

|Gril p. 178

|

• 0∈ S
• 0 ≠ x ≤_R y, z   ⇒   y ≤_L z or z ≤_R y

|

|Gril p. 178

|

• 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

|

• a ∈ bS¹   ⇒   aⁿ ∈ b²S¹ for some n.

|

|Nagy p. 35

|

• a ∈ S¹b   ⇒   aⁿ ∈ S¹b² for some n.

|

|Nagy p. 35

|

• a ∈ S¹b S¹   ⇒   aⁿ ∈ S¹b²S¹ for some positive integer n

|

|Nagy p. 35

|

• D_a = S

|

|C&P p. 49

|

• 0 ∈ S
• S - {0} is a D-class of S.

|

|C&P p. 76

|

• 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

|

• 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_a = S

|

|C&P p. 49

|

• Let J(a) = S¹aS¹, I(a) = J(a) − J_a. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.

|

|C&P p. 71–75

|

• J_a = S. (There exists no A ⊆ S, A ≠ S such that SA ⊆ A and AS ⊆ A.),
• equivalently, for finite semigroup: $a^\{\backslash omega\}a=a$ and $(aba)^\backslash omega=a^\backslash omega$.

|

• Finite

|

• C&P p. 5
• Higg p. 16
• Pin pp. 151, 158

|

• 0 ∈ S
• S² ≠ 0
• If A ⊆ S is such that AS ⊆ A and SA ⊆ A then A = 0.

|

|C&P p. 67

|

• 0 ∈ S
• S² ≠ 0
• If A ⊆ S is such that SA ⊆ A then A = 0.

|

|C&P p. 67

|

• 0 ∈ S
• S² ≠ 0
• If A ⊆ S is such that AS ⊆ A then A = 0.

|

|C&P p. 67

|

• S = { w, w², w³, ... } for some w in S

|

• Not infinite
• Not finite

|C&P p. 19

|

• { a, a², a³, ... } is a finite set.

|

• Not infinite
• Finite

|C&P p. 20

|

• 1 ∈ S
• S admits the presentation $\backslash langle\; x,y\backslash mid\; xy=1\backslash rangle$.

|

|C&P p. 43–46

|

• Set of all mappings of X into itself with composition of mappings as binary operation.

|

|C&P p. 2

|

• A band such that aba = a
• Equivalently abc = ac

|

• Infinite
• Finite

|Fennemore

|

• Whenever three of ax, ay, bx, by are equal, all four are equal.

|

|C&P p. 97

|

• The semigroup of one-to-one partial transformations of X.

|

|C&P p. 29

|

• 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

|

• Set of finite sequences of elements of X with the operation
  ( x₁, ..., x_m ) ( y₁, ..., y_n ) = ( x₁, ..., x_m, y₁, ..., y_n )

|

|Gril p. 18

|

• 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 of a vector space V over a field F under composition of functions.

|

|C&P p.57

|

• Set of all binary relations on X under composition

|

|C&P p.13

|

• 0 ∈ S ⊆ N = { 0,1,2, ... } under + .
• N - S is finite

|

|Delg

|

• There exists a unary operation a → a* in S such that a** = a and (ab)* = b*a*.

|

|Howi

|

• Semigroup of one-to-one transformations f of X such that X − f ( X ) is infinite.

|

|C&P II Ch.8

|

• There exists a unary operation a → a’ in S such that ( a’)’ = a.

|

|Howi p.102

|

• There exists a unary operation a → a’ in S such that ( a’)’ = a and aa’a = a.

|

|Howi p.102

|

• A regular semigroup generated by its idempotents.

|

|Howi p.230

|

• 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

|

|

• A semigroup which is also a topological space. Such that the semigroup product is continuous.

|

• Not applicable

|Pin p. 130

|

• The smallest finite monoid which can recognize a subset of another semigroup.

|

|Pin p. 14

|

• R-trivial. That is, each R-equivalence class is trivial.
• Equivalently, for finite semigroup: $(ab)^\backslash omega\; a=(ab)^\backslash omega$.

|

• Finite

|Pin p. 158

|

• L-trivial. That is, each L-equivalence class is trivial.
• Equivalently, for finite monoids, $b(ab)^\backslash omega=(ab)^\backslash omega$.

|

• Finite

|Pin p. 158

|

• 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

|

• R-trivial. That is, each R-equivalence class is trivial.
• Equivalently, for finite monoids: aba = ab.

|

• Finite

|Pin p. 158

|

• L-trivial. That is, each L-equivalence class is trivial.
• Equivalently, for finite monoids: aba = ba.

|

• Finite

|Pin p. 158

|

• Equivalently, for finite monoids: $(a^\backslash omega\; a^\backslash omega\; a^\backslash omega)^\backslash omega=a^\backslash omega$.
• 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 $B^1\_2$ does not divide $S\backslash times\; S$

|

• Finite

|Pin pp. 154, 155, 158

|

• 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 $aa^\backslash omega=a^\backslash omega$
• Equivalently, e≤_Ja implies eae = e
• Equivalently, e≤_Jf implies efe = e.

|

• Finite

|Pin p. 156, 158

|

• e: eS = e,
• Equivalently, I is a left zero semigroup equal to E,
• Equivalently, for finite semigroup: I is a left zero semigroup equals $S^\{|S$

|

|+List of special classes of ordered semigroups

|

• 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

|

• Nilpotent finite semigroups, with $a\backslash le\; b^\backslash omega$

|

• Finite

|Pin pp. 157, 158

|

• Nilpotent finite semigroups, with $b^\backslash omega\backslash le\; a$

|

• Finite

|Pin pp. 157, 158

|

• Semilattices with $1\backslash le\; a$

|

• Finite

|Pin pp. 157, 158

|

• Semilattices with $a\backslash le\; 1$

|

• Finite

|Pin pp. 157, 158

|

• Finite semigroups satisfying $a^\backslash omega\backslash le\; a^\backslash omega\; ba^\backslash omega$

|

• Finite

|Pin pp. 157, 158

| [C&P]

|

| {{Anchor|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]  

|

|{{Anchor|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] 

|

| {{Anchor|Chen}}Hui Chen (2006), "Construction of a kind of abundant semigroups", Mathematical Communications (11), 165–171 (Accessed on 25 April 2009)

|[Delg]

|

|{{Anchor|Delg}}M. Delgado, et al., Numerical semigroups, [http://www.gap-system.org/Manuals/pkg/numericalsgps/doc/manual.pdf] (Accessed on 27 April 2009)

|[Edwa]

|

|{{Anchor|Edwa}}P. M. Edwards (1983), "Eventually regular semigroups", Bulletin of Australian Mathematical Society 28, 23–38

|[Gril]

|

|{{Anchor|Gril}}P. A. Grillet (1995). Semigroups. CRC Press. {{isbn|978-0-8247-9662-4}}

|[Hari]

|

|{{Anchor|Hari}}K. S. Harinath (1979), "Some results on k-regular semigroups", Indian Journal of Pure and Applied Mathematics 10(11), 1422–1431

|[Howi]

|

|{{Anchor|Howi}}J. M. Howie (1995), Fundamentals of Semigroup Theory, Oxford University Press

|[Nagy]

|

|{{Anchor|Nagy}}Attila Nagy (2001). Special Classes of Semigroups. Springer. {{isbn|978-0-7923-6890-8}}

|[Pet]

|

|{{Anchor|Pet}} M. Petrich, N. R. Reilly (1999). Completely regular semigroups. John Wiley & Sons. {{isbn|978-0-471-19571-9}}

|[Shum]    

|

|{{Anchor|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]

|

|{{Anchor|Tvm}}Proceedings of the International Symposium on Theory of Regular Semigroups and Applications, University of Kerala, Thiruvananthapuram, India, 1986

|[Kela]

|

|{{Anchor|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]

|

|{{Anchor|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]

|

|{{Anchor|Higg}} {{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5}}

|[Pin]

|

|{{Anchor|Pin}}{{cite book|last1=Pin|first1=Jean-Éric|authorlink = Jean-Éric Pin|title=Mathematical Foundations of Automata Theory|date=2016-11-30|url=http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf}}

|[Fennemore]

|

|{{Anchor|Fennemore}}{{citation

| last = Fennemore | first = Charles

| doi = 10.1007/BF02573031

| issue = 1

| journal = Semigroup Forum

| pages = 172–179

| title = All varieties of bands

| volume = 1

| year = 1970}}