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 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.
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 |
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. |
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.
Terminology | Defining property | Variety of finite semigroup | Reference(s) |
---|---|---|---|
Finite semigroup |
|
|
|
Empty semigroup |
|
No | |
Trivial semigroup |
|
|
|
Monoid |
|
No | Gril p. 3 |
Band (Idempotent semigroup) |
|
|
C&P p. 4 |
Rectangular band |
|
|
Fennemore |
Semilattice | A commutative band, that is:
|
|
|
Commutative semigroup |
|
|
C&P p. 3 |
Archimedean commutative semigroup |
|
C&P p. 131 | |
Nowhere commutative semigroup |
|
C&P p. 26 | |
Left weakly commutative |
|
Nagy p. 59 | |
Right weakly commutative |
|
Nagy p. 59 | |
Weakly commutative | Left and right weakly commutative. That is:
|
Nagy p. 59 | |
Conditionally commutative semigroup |
|
Nagy p. 77 | |
R-commutative semigroup |
|
Nagy p. 69–71 | |
RC-commutative semigroup |
|
Nagy p. 93–107 | |
L-commutative semigroup |
|
Nagy p. 69–71 | |
LC-commutative semigroup |
|
Nagy p. 93–107 | |
H-commutative semigroup |
|
Nagy p. 69–71 | |
Quasi-commutative semigroup |
|
Nagy p. 109 | |
Right commutative semigroup |
|
Nagy p. 137 | |
Left commutative semigroup |
|
Nagy p. 137 | |
Externally commutative semigroup |
|
Nagy p. 175 | |
Medial semigroup |
|
Nagy p. 119 | |
E-k semigroup (k fixed) |
|
|
Nagy p. 183 |
Exponential semigroup |
|
|
Nagy p. 183 |
WE-k semigroup (k fixed) |
|
Nagy p. 199 | |
Weakly exponential semigroup |
|
Nagy p. 215 | |
Right cancellative semigroup |
|
C&P p. 3 | |
Left cancellative semigroup |
|
C&P p. 3 | |
Cancellative semigroup | Left and right cancellative semigroup, that is
|
C&P p. 3 | |
''E''-inversive semigroup (E-dense semigroup) |
|
C&P p. 98 | |
Regular semigroup |
|
C&P p. 26 | |
Regular band |
|
|
Fennemore |
Intra-regular semigroup |
|
C&P p. 121 | |
Left regular semigroup |
|
C&P p. 121 | |
Left-regular band |
|
|
Fennemore |
Right regular semigroup |
|
C&P p. 121 | |
Right-regular band |
|
|
Fennemore |
Completely regular semigroup |
|
Gril p. 75 | |
(inverse) Clifford semigroup |
|
|
Petrich p. 65 |
k-regular semigroup (k fixed) |
|
Hari | |
Eventually regular semigroup (π-regular semigroup, Quasi regular semigroup) |
|
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) |
|
Kela Gril p. 110 Higg p. 4 | |
Primitive semigroup |
|
C&P p. 26 | |
Unit regular semigroup |
|
Tvm | |
Strongly unit regular semigroup |
|
Tvm | |
Orthodox semigroup |
|
Gril p. 57 Howi p. 226 | |
Inverse semigroup |
|
C&P p. 28 | |
Left inverse semigroup (R-unipotent) |
|
Gril p. 382 | |
Right inverse semigroup (L-unipotent) |
|
Gril p. 382 | |
Locally inverse semigroup (Pseudoinverse semigroup) |
|
Gril p. 352 | |
M-inversive semigroup |
|
C&P p. 98 | |
Pseudoinverse semigroup (Locally inverse semigroup) |
|
Gril p. 352 | |
Abundant semigroup |
|
Chen | |
Rpp-semigroup (Right principal projective semigroup) |
|
Shum | |
Lpp-semigroup (Left principal projective semigroup) |
|
Shum | |
Null semigroup (Zero semigroup) |
|
|
C&P p. 4 |
Left zero semigroup |
|
|
C&P p. 4 |
Left zero band | A left zero semigroup which is a band. That is:
|
|
|
Right zero semigroup |
|
|
C&P p. 4 |
Right zero band | A right zero semigroup which is a band. That is:
|
|
Fennemore |
Unipotent semigroup |
|
|
C&P p. 21 |
Left reductive semigroup |
|
C&P p. 9 | |
Right reductive semigroup |
|
C&P p. 4 | |
Reductive semigroup |
|
C&P p. 4 | |
Separative semigroup |
|
C&P p. 130–131 | |
Reversible semigroup |
|
C&P p. 34 | |
Right reversible semigroup |
|
C&P p. 34 | |
Left reversible semigroup |
|
C&P p. 34 | |
Aperiodic semigroup |
|
||
ω-semigroup |
|
Gril p. 233–238 | |
Left Clifford semigroup (LC-semigroup) |
|
Shum | |
Right Clifford semigroup (RC-semigroup) |
|
Shum | |
Orthogroup |
|
Shum | |
Complete commutative semigroup |
|
Gril p. 110 | |
Nilsemigroup (Nilpotent semigroup) |
|
|
|
Elementary semigroup |
|
Gril p. 111 | |
E-unitary semigroup |
|
Gril p. 245 | |
Finitely presented semigroup |
|
Gril p. 134 | |
Fundamental semigroup |
|
Gril p. 88 | |
Idempotent generated semigroup |
|
Gril p. 328 | |
Locally finite semigroup |
|
|
Gril p. 161 |
N-semigroup |
|
Gril p. 100 | |
L-unipotent semigroup (Right inverse semigroup) |
|
Gril p. 362 | |
R-unipotent semigroup (Left inverse semigroup) |
|
Gril p. 362 | |
Left simple semigroup |
|
Gril p. 57 | |
Right simple semigroup |
|
Gril p. 57 | |
Subelementary semigroup |
|
Gril p. 134 | |
Symmetric semigroup (Full transformation semigroup) |
|
C&P p. 2 | |
Weakly reductive semigroup |
|
C&P p. 11 | |
Right unambiguous semigroup |
|
Gril p. 170 | |
Left unambiguous semigroup |
|
Gril p. 170 | |
Unambiguous semigroup |
|
Gril p. 170 | |
Left 0-unambiguous |
|
Gril p. 178 | |
Right 0-unambiguous |
|
Gril p. 178 | |
0-unambiguous semigroup |
|
Gril p. 178 | |
Left Putcha semigroup |
|
Nagy p. 35 | |
Right Putcha semigroup |
|
Nagy p. 35 | |
Putcha semigroup |
|
Nagy p. 35 | |
Bisimple semigroup (D-simple semigroup) |
|
C&P p. 49 | |
0-bisimple semigroup |
|
C&P p. 76 | |
Completely simple semigroup |
|
C&P p. 76 | |
Completely 0-simple semigroup |
|
C&P p. 76 | |
D-simple semigroup (Bisimple semigroup) |
|
C&P p. 49 | |
Semisimple semigroup |
|
C&P p. 71–75 | |
: Simple semigroup |
|
|
|
0-simple semigroup |
|
C&P p. 67 | |
Left 0-simple semigroup |
|
C&P p. 67 | |
Right 0-simple semigroup |
|
C&P p. 67 | |
Cyclic semigroup (Monogenic semigroup) |
|
|
C&P p. 19 |
Periodic semigroup |
|
|
C&P p. 20 |
Bicyclic semigroup |
|
C&P p. 43–46 | |
Full transformation semigroup TX (Symmetric semigroup) |
|
C&P p. 2 | |
Rectangular band |
|
|
Fennemore |
Rectangular semigroup |
|
C&P p. 97 | |
Symmetric inverse semigroup IX |
|
C&P p. 29 | |
Brandt semigroup |
|
C&P p. 101 | |
Free semigroup FX |
|
Gril p. 18 | |
Rees matrix semigroup |
|
C&P p.88 | |
Semigroup of linear transformations |
|
C&P p.57 | |
Semigroup of binary relations BX |
|
C&P p.13 | |
Numerical semigroup |
|
Delg | |
Semigroup with involution (*-semigroup) |
|
Howi | |
Baer–Levi semigroup |
|
C&P II Ch.8 | |
U-semigroup |
|
Howi p.102 | |
I-semigroup |
|
Howi p.102 | |
Semiband |
|
Howi p.230 | |
Group |
|
|
|
Topological semigroup |
|
|
Pin p. 130 |
Syntactic semigroup |
|
Pin p. 14 | |
: the R-trivial monoids |
|
|
Pin p. 158 |
: the L-trivial monoids |
|
|
Pin p. 158 |
: the J-trivial monoids |
|
|
Pin p. 158 |
: idempotent and R-trivial monoids |
|
|
Pin p. 158 |
: idempotent and L-trivial monoids |
|
|
Pin p. 158 |
: Semigroup whose regular D are semigroup |
|
|
Pin pp. 154, 155, 158 |
: Semigroup whose regular D are aperiodic semigroup |
|
|
Pin p. 156, 158 |
/: Lefty trivial semigroup |
|
|
Pin pp. 149, 158 |
/: Right trivial semigroup |
|
|
Pin pp. 149, 158 |
: Locally trivial semigroup |
|
|
Pin pp. 150, 158 |
: Locally groups |
|
|
Pin pp. 151, 158 |
Terminology | Defining property | Variety | Reference(s) |
---|---|---|---|
Ordered semigroup |
|
|
Pin p. 14 |
|
|
Pin pp. 157, 158 | |
|
|
Pin pp. 157, 158 | |
|
|
Pin pp. 157, 158 | |
|
|
Pin pp. 157, 158 | |
locally positive J-trivial semigroup |
|
|
Pin pp. 157, 158 |
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, [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. | |
[Pin] | Pin, Jean-Éric (2016-11-30). Mathematical Foundations of Automata Theory (PDF). | |
[Fennemore] | Fennemore, Charles (1970), "All varieties of bands", Semigroup Forum, 1 (1): 172–179, doi:10.1007/BF02573031 |