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
aL b
aR b
aH b
aJ b
S1aS1b
aS1bS1
S1aS1b and aS1bS1
S1aS1S1bS1
L, R, H, D, J Green's relations
La, Ra, Ha, Da, Ja Green classes containing a
${\displaystyle x^{\omega }}$ The only power of x which is idempotent. This element exists, assimng the semigroup is (locally) finite. See variety of finite semigroups for more information about this notation.
${\displaystyle |X|}$ 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 state whether this set of semigroup form a variety. And whether the set of finite semigroups of this special classes forms a variety of finite semigroups. Not that if this set is a variety, its set of finite element is automatically a variety of finite semigroup.

List of special classes of semigroups
Terminology Defining property Variety of finite semigroup Reference(s)
Finite semigroup
• Not infinite
• Finite
Empty semigroup
• S = ${\displaystyle \emptyset }$
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
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 = a   ⇒   b = c
C&P p. 3
''E''-inversive semigroup (E-dense semigroup)
• There exists x such that axE.
C&P p. 98
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: ${\displaystyle a^{\omega }b=ba^{\omega }}$
• 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 0e 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 ff = 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 semigroup
• The classes L*a and R*a, where a L* b if ac = adbc = bd and a R* b if ca = dacb = db, contain idempotents.
Chen
Rpp-semigroup
(Right principal projective semigroup)
• The class L*a, where a L* b if ac = adbc = bd, contains at least one idempotent.
Shum
Lpp-semigroup
(Left principal projective semigroup)
• The class R*a, where a R* b if ca = dacb = 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
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
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
• SaSb ≠ Ø
• aSbS ≠ Ø
C&P p. 34
Right reversible semigroup
• SaSb ≠ Ø
C&P p. 34
Left reversible semigroup
• aSbS ≠ 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, ${\displaystyle a^{\omega }a=a^{\omega }}$.
ω-semigroup
• E is countable descending chain under the order aH b
Gril p. 233–238
Left Clifford semigroup
(LC-semigroup)
• aSSa
Shum
Right Clifford semigroup
(RC-semigroup)
• SaaS
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, ${\displaystyle yx^{\omega }=x^{\omega }=x^{\omega }y}$.
• Finite
Elementary semigroup
• ab = ba
• S is of the form GN 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   ⇒   aE
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 = CN 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, yR z then xR y or yR x.
Gril p. 170
Left unambiguous semigroup
• If x, yL z then xL y or yL x.
Gril p. 170
Unambiguous semigroup
• If x, yR z then xR y or yR x.
• If x, yL z then xL y or yL x.
Gril p. 170
Left 0-unambiguous
• 0∈ S
• 0 ≠ xL y, z   ⇒   yL z or zL y
Gril p. 178
Right 0-unambiguous
• 0∈ S
• 0 ≠ xR y, z   ⇒   yL z or zR y
Gril p. 178
0-unambiguous semigroup
• 0∈ S
• 0 ≠ xL y, z   ⇒   yL z or zL y
• 0 ≠ xR y, z   ⇒   yL z or zR y
Gril p. 178
Left Putcha semigroup
• abS1   ⇒   anb2S1 for some n.
Nagy p. 35
Right Putcha semigroup
• aS1b   ⇒   anS1b2 for some n.
Nagy p. 35
Putcha semigroup
• aS1b S1   ⇒   anS1b2S1 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 AS, AS such that SAA and ASA.
• 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 AS is such that ASA and SAA 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) = S1aS1, I(a) = J(a) − Ja. Each Rees factor semigroup J(a)/I(a) is 0-simple or simple.
C&P p. 71–75
${\displaystyle \mathbf {CS} }$: Simple semigroup
• Ja = S. (There exists no AS, AS such that SAA and ASA.),
• equivalently, for finite semigroup: ${\displaystyle a^{\omega }a=a}$ and ${\displaystyle (aba)^{\omega }=a^{\omega }}$.
• Finite
0-simple semigroup
• 0 ∈ S
• S2 ≠ 0
• If AS is such that ASA and SAA then A = 0.
C&P p. 67
Left 0-simple semigroup
• 0 ∈ S
• S2 ≠ 0
• If AS is such that SAA then A = 0.
C&P p. 67
Right 0-simple semigroup
• 0 ∈ S
• S2 ≠ 0
• If AS is such that ASA 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 ${\displaystyle \langle x,y\mid xy=1\rangle }$.
C&P p. 43–46
Full transformation semigroup TX
(Symmetric semigroup)
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
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 × IG0 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 C&P p.57
Semigroup of binary relations BX
C&P p.13
Numerical semigroup
• 0 ∈ SN = { 0,1,2, ... } under + .
• N - S is finite
Delg
Semigroup with involution
(*-semigroup)
• There exists a unary operation aa* 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 Xf ( X ) is infinite.
C&P II Ch.8
U-semigroup
• There exists a unary operation aa’ in S such that ( a’)’ = a.
Howi p.102
I-semigroup
• There exists a unary operation aa’ in S such that ( a’)’ = a and aaa = 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
${\displaystyle \mathbf {R} }$: the R-trivial monoids
• R-trivial. That is, each R-equivalence class is trivial.
• Equivalently, for finite semigroup: ${\displaystyle (ab)^{\omega }a=(ab)^{\omega }}$.
• Finite
Pin p. 158
${\displaystyle \mathbf {L} }$: the L-trivial monoids
• L-trivial. That is, each L-equivalence class is trivial.
• Equivalently, for finite monoids, ${\displaystyle b(ab)^{\omega }=(ab)^{\omega }}$.
• Finite
Pin p. 158
${\displaystyle \mathbf {J} }$: 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-trivia.
• Finite
Pin p. 158
${\displaystyle \mathbf {R_{1}} }$: 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
${\displaystyle \mathbf {L_{1}} }$: 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
${\displaystyle \mathbb {D} \mathbf {S} }$: Semigroup whose regular D are semigroup
• Equivalently, for finite monoids: ${\displaystyle (a^{\omega }a^{\omega }a^{\omega })^{\omega }=a^{\omega }}$.
• Equivalently, regular H-classes are groups,
• Equivalently, vJa implies v R va and v L av
• Equivalently, for each idempotent e, the set of a such that eJa 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 ${\displaystyle B_{2}^{1}}$ does not divide ${\displaystyle S\times S}$
• Finite
Pin pp. 154, 155, 158
${\displaystyle \mathbb {D} \mathbf {A} }$: 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 ${\displaystyle aa^{\omega }=a^{\omega }}$
• Equivalently, eJa implies eae = e
• Equivalently, eJf implies efe = e.
• Finite
Pin p. 156, 158
${\displaystyle \ell \mathbf {1} }$/${\displaystyle \mathbf {K} }$: 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 ${\displaystyle S^{|S|}}$,
• Equivalently, for finite semigroup: ${\displaystyle a_{1}\dots a_{n}y=a_{1}\dots a_{n}}$,
• Equivalently, for finite semigroup: ${\displaystyle a^{\omega }b=a^{\omega }}$.
• Finite
Pin pp. 149, 158
${\displaystyle \mathbf {r1} }$/${\displaystyle \mathbf {D} }$: 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 ${\displaystyle S^{|S|}}$,
• Equivalently, for finite semigroup: ${\displaystyle ba_{1}\dots a_{n}=a_{1}\dots a_{n}}$,
• Equivalently, for finite semigroup: ${\displaystyle ba^{\omega }=a^{\omega }}$.
• Finite
Pin pp. 149, 158
${\displaystyle \mathbb {L} \mathbf {1} }$: Locally trivial semigroup
• eSe = e,
• Equivalently, I is equal to E,
• Equivalently, eaf = ef,
• Equivalently, for finite semigroup: ${\displaystyle ya_{1}\dots a_{n}=a_{1}\dots a_{n}}$,
• Equivalently, for finite semigroup: ${\displaystyle a_{1}\dots a_{n}ya_{1}\dots a_{n}=a_{1}\dots a_{n}}$,
• Equivalently, for finite semigroup: ${\displaystyle a^{\omega }ba^{\omega }=a^{\omega }}$.
• Finite
Pin pp. 150, 158
${\displaystyle \mathbb {L} \mathbf {G} }$: Locally groups
• eSe is a group,
• Equivalently, EI,
• Equivalently, for finite semigroup: ${\displaystyle (a^{\omega }ba^{\omega })^{\omega }=a^{\omega }}$.
• 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 ${\displaystyle \mathbf {N} ^{+}}$ Nilpotent finite semigroups, with ${\displaystyle a\leq b^{\omega }}$ Finite Pin pp. 157, 158 ${\displaystyle \mathbf {N} ^{-}}$ Nilpotent finite semigroups, with ${\displaystyle b^{\omega }\leq a}$ Finite Pin pp. 157, 158 ${\displaystyle \mathbf {J} _{1}^{+}}$ Semilattices with ${\displaystyle 1\leq a}$ Finite Pin pp. 157, 158 ${\displaystyle \mathbf {J} _{1}^{-}}$ Semilattices with ${\displaystyle a\leq 1}$ Finite Pin pp. 157, 158 ${\displaystyle \mathbb {L} \mathbf {J} _{1}^{+}}$ locally positive J-trivial semigroup Finite semigroups satisfying ${\displaystyle a^{\omega }\leq a^{\omega }ba^{\omega }}$ Finite Pin pp. 157, 158

References

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