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[edit]
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 
S^{1}  S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S 
a ≤_{L} b a ≤_{R} b a ≤_{H} b a ≤_{J} b 
S^{1}a ⊆ S^{1}b aS^{1} ⊆ bS^{1} S^{1}a ⊆ S^{1}b and aS^{1} ⊆ bS^{1} S^{1}aS^{1} ⊆ S^{1}bS^{1} 
L, R, H, D, J  Green's relations 
L_{a}, R_{a}, H_{a}, D_{a}, J_{a}  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[edit]
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  
Rcommutative semigroup 

Nagy p. 69–71  
RCcommutative semigroup 

Nagy p. 93–107  
Lcommutative semigroup 

Nagy p. 69–71  
LCcommutative semigroup 

Nagy p. 93–107  
Hcommutative semigroup 

Nagy p. 69–71  
Quasicommutative 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  
Ek semigroup (k fixed) 


Nagy p. 183 
Exponential semigroup 


Nagy p. 183 
WEk 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 (Edense semigroup) 

C&P p. 98  
Regular semigroup 

C&P p. 26  
Regular band 


Fennemore 
Intraregular semigroup 

C&P p. 121  
Left regular semigroup 

C&P p. 121  
Leftregular band 


Fennemore 
Right regular semigroup 

C&P p. 121  
Rightregular band 


Fennemore 
Completely regular semigroup 

Gril p. 75  
(inverse) Clifford semigroup 


Petrich p. 65 
kregular semigroup (k fixed) 

Hari  
Eventually regular semigroup (πregular semigroup, Quasi regular semigroup) 

Edwa Shum Higg p. 49  
Quasiperiodic semigroup, epigroup, groupbound semigroup, completely (or strongly) πregular 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 (Runipotent) 

Gril p. 382  
Right inverse semigroup (Lunipotent) 

Gril p. 382  
Locally inverse semigroup (Pseudoinverse semigroup) 

Gril p. 352  
Minversive semigroup 

C&P p. 98  
Pseudoinverse semigroup (Locally inverse semigroup) 

Gril p. 352  
Abundant semigroup 

Chen  
Rppsemigroup (Right principal projective semigroup) 

Shum  
Lppsemigroup (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 


ωsemigroup 

Gril p. 233–238  
Left Clifford semigroup (LCsemigroup) 

Shum  
Right Clifford semigroup (RCsemigroup) 

Shum  
Orthogroup 

Shum  
Complete commutative semigroup 

Gril p. 110  
Nilsemigroup (Nilpotent semigroup) 



Elementary semigroup 

Gril p. 111  
Eunitary 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 
Nsemigroup 

Gril p. 100  
Lunipotent semigroup (Right inverse semigroup) 

Gril p. 362  
Runipotent 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 0unambiguous 

Gril p. 178  
Right 0unambiguous 

Gril p. 178  
0unambiguous semigroup 

Gril p. 178  
Left Putcha semigroup 

Nagy p. 35  
Right Putcha semigroup 

Nagy p. 35  
Putcha semigroup 

Nagy p. 35  
Bisimple semigroup (Dsimple semigroup) 

C&P p. 49  
0bisimple semigroup 

C&P p. 76  
Completely simple semigroup 

C&P p. 76  
Completely 0simple semigroup 

C&P p. 76  
Dsimple semigroup (Bisimple semigroup) 

C&P p. 49  
Semisimple semigroup 

C&P p. 71–75  
: Simple semigroup 



0simple semigroup 

C&P p. 67  
Left 0simple semigroup 

C&P p. 67  
Right 0simple 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 T_{X} (Symmetric semigroup) 

C&P p. 2  
Rectangular band 


Fennemore 
Rectangular semigroup 

C&P p. 97  
Symmetric inverse semigroup I_{X} 

C&P p. 29  
Brandt semigroup 

C&P p. 101  
Free semigroup F_{X} 

Gril p. 18  
Rees matrix semigroup 

C&P p.88  
Semigroup of linear transformations 

C&P p.57  
Semigroup of binary relations B_{X} 

C&P p.13  
Numerical semigroup 

Delg  
Semigroup with involution (*semigroup) 

Howi  
Baer–Levi semigroup 

C&P II Ch.8  
Usemigroup 

Howi p.102  
Isemigroup 

Howi p.102  
Semiband 

Howi p.230  
Group 



Topological semigroup 


Pin p. 130 
Syntactic semigroup 

Pin p. 14  
: the Rtrivial monoids 


Pin p. 158 
: the Ltrivial monoids 


Pin p. 158 
: the Jtrivial monoids 


Pin p. 158 
: idempotent and Rtrivial monoids 


Pin p. 158 
: idempotent and Ltrivial 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 Jtrivial semigroup 


Pin pp. 157, 158 
References[edit]
[C&P]  A. H. Clifford, G. B. Preston (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. ISBN 9780821802724  
[C&P II]  A. H. Clifford, G. B. Preston (1967). The Algebraic Theory of Semigroups Vol. II (Second Edition). American Mathematical Society. ISBN 0821802720  
[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 9780824796624  
[Hari]  K. S. Harinath (1979), "Some results on kregular 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 9780792368908  
[Pet]  M. Petrich, N R Reilly (1999). Completely regular semigroups. John Wiley & Sons. ISBN 9780471195719  
[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 9812790004 (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), 327350 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 9783110152487.  
[Higg]  Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. ISBN 9780198535775.  
[Pin]  Pin, JeanÉric (20161130). Mathematical Foundations of Automata Theory (PDF).  
[Fennemore]  Fennemore, Charles (1970), "All varieties of bands", Semigroup Forum, 1 (1): 172–179, doi:10.1007/BF02573031 