Special classes of semigroups

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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[edit]

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
S1 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
aL b
aR b
aH b
S1aS1b
aS1bS1
S1aS1b and aS1bS1

List of special classes of semigroups[edit]

List of special classes of semigroups
Terminology Defining property Reference(s)
Finite semigroup
Empty semigroup
  • S = \emptyset
Trivial semigroup
  • Cardinality of S is 1.
Monoid
  • 1 ∈ S
Gril p. 3
Band
(Idempotent semigroup)
  • a2 = a
C&P p. 4
Semilattice
  • a2 = 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 = xbk.
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 = akbk
Nagy p. 183
Exponential semigroup
  • (ab)m = ambm 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 = akbk (ab)j = (ab)jakbk
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 axE.
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 xa2y = a.
C&P p. 121
Left regular semigroup
  • There exists x such that xa2 = a.
C&P p. 121
Right regular semigroup
  • There exists x such that a2x = 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 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 semigroups
  • 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
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 = 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
KKM p. 29
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
LC-semigroup
(Left Clifford semigroup)
  • aSSa
Shum
RC-semigroup
(Right Clifford 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
  • 0 ∈ S
  • ak = 0 for some k.
Gril p. 99
Elementary semigroup
  • ab = ba
  • S = GN 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   ⇒   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.
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
Simple semigroup
  • Ja = S. (There exists no AS, AS such that SAA and ASA.)
C&P p. 5
Higg p. 16
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
C&P p. 19
Monogenic semigroup
(Cyclic semigroup)
  • S = { w, w2, w3, ... } for some w in S
C&P p. 19
Periodic semigroup
  • { a, a2, a3, ... } 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)
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 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
*-semigroup
(Semigroup with involution)
  • 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.

References[edit]

[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, [1] (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.