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 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
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 
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 
S^{1}  S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S 
L, R, H, D, J  Green's relations 
L_{a}, R_{a}, H_{a}, D_{a}, J_{a}  Green classes containing a 
a ≤_{L} b a ≤_{R} b a ≤_{H} b 
S^{1}a ⊆ S^{1}b aS^{1} ⊆ bS^{1} S^{1}a ⊆ S^{1}b and aS^{1} ⊆ bS^{1} 
List of special classes of semigroups
Terminology  Defining property  Reference(s) 

Finite semigroup 


Empty semigroup 


Trivial semigroup 


Monoid 

Gril p. 3 
Band (Idempotent semigroup) 

C&P p. 4 
Semilattice 

C&P p. 24 
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 

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 
Cancellative semigroup 

C&P p. 3 
Right cancellative semigroup 

C&P p. 3 
Left cancellative semigroup 

C&P p. 3 
Einversive semigroup 

C&P p. 98 
Regular semigroup 

C&P p. 26 
Intraregular semigroup 

C&P p. 121 
Left regular semigroup 

C&P p. 121 
Right regular semigroup 

C&P p. 121 
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 semigroups 

Chen 
Rppsemigroup (Right principal projective semigroup) 

Shum 
Lppsemigroup (Left principal projective semigroup) 

Shum 
Null semigroup (Zero semigroup) 

C&P p. 4 
Zero semigroup (Null semigroup) 

C&P p. 4 
Left zero semigroup 

C&P p. 4 
Right zero semigroup 

C&P p. 4 
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 

KKM p. 29 
ωsemigroup 

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

Shum 
Right Clifford semigroup (RCsemigroup) 

Shum 
LCsemigroup (Left Clifford semigroup) 

Shum 
RCsemigroup (Right Clifford semigroup) 

Shum 
Orthogroup 

Shum 
Complete commutative semigroup 

Gril p. 110 
Nilsemigroup 

Gril p. 99 
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 

C&P p. 5 Higg p. 16 
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 
Monogenic semigroup (Cyclic 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 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 
*semigroup (Semigroup with involution) 

Howi 
Baer–Levi semigroup 

C&P II Ch.8 
Usemigroup 

Howi p.102 
Isemigroup 

Howi p.102 
Semiband 

Howi p.230 
Group 

References
[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, (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. 