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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
is a
nonempty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
together with an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. A special class of semigroups is a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
of
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
s satisfying additional property (philosophy), 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 set, 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 set (mathematics), sets are not assumed to carry any other mathematical structures like Partial order, order or topology. As in any algebraic theory, one of the main problems of the theory of semigroups is the classification theorems, 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 (mathematics), 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.
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 (universal algebra), 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.


References

{, , -valign="top" , [C&P] , , A. H. Clifford, G. B. Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). American Mathematical Society. , -valign="top" , [C&P II]   , , A. H. Clifford, G. B. Preston (1967). ''The Algebraic Theory of Semigroups Vol. II'' (Second Edition). American Mathematical Society. , -valign="top" , [Chen]  , , Hui Chen (2006), "Construction of a kind of abundant semigroups", ''Mathematical Communications'' (11), 165–171 (Accessed on 25 April 2009) , -valign="top" , [Delg] , , M. Delgado, ''et al.'', ''Numerical semigroups''

(Accessed on 27 April 2009) , -valign="top" , [Edwa] , , P. M. Edwards (1983), "Eventually regular semigroups", ''Bulletin of Australian Mathematical Society'' 28, 23–38 , -valign="top" , [Gril] , , P. A. Grillet (1995). ''Semigroups''. CRC Press. , -valign="top" , [Hari] , , K. S. Harinath (1979), "Some results on ''k''-regular semigroups", ''Indian Journal of Pure and Applied Mathematics'' 10(11), 1422–1431 , -valign="top" , [Howi] , , John Mackintosh Howie, J. M. Howie (1995), ''Fundamentals of Semigroup Theory'', Oxford University Press , -valign="top" , [Nagy] , , Attila Nagy (2001). ''Special Classes of Semigroups''. Springer Science+Business Media, Springer. , -valign="top" , [Pet] , , M. Petrich, N. R. Reilly (1999). ''Completely regular semigroups''. John Wiley & Sons. , -valign="top" , [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, (pp. 303–334) , -valign="top" , [Tvm] , , ''Proceedings of the International Symposium on Theory of Regular Semigroups and Applications'', University of Kerala, Thiruvananthapuram, India, 1986 , -valign="top" , [Kela] , , A. V. Kelarev, ''Applications of epigroups to graded ring theory'', Semigroup Forum, Volume 50, Number 1 (1995), 327-350 , -valign="top" , [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, . , -valign="top" , [Higg] , , , -valign="top" , [Pin] , , , -valign="top" , [Fennemore] , , {{citation , last = Fennemore , first = Charles , doi = 10.1007/BF02573031 , issue = 1 , journal = Semigroup Forum , pages = 172–179 , title = All varieties of bands , volume = 1 , year = 1970 , -valign="top" Algebraic structures Semigroup theory