Commutative semigroup
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In mathematics, a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
is a
nonempty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
together with an associative binary operation. A special class of semigroups is a class of
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s satisfying additional
properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy an ...
or conditions. Thus the class of
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
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
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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 group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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 Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
. 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" , &P, , A. H. Clifford, G. B. Preston (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition).
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. , -valign="top" , &P II  , , A. H. Clifford, G. B. Preston (1967). ''The Algebraic Theory of Semigroups Vol. II'' (Second Edition).
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. , -valign="top" ,
hen Hen commonly refers to a female animal: a female chicken, other gallinaceous bird, any type of bird in general, or a lobster. It is also a slang term for a woman. Hen or Hens may also refer to: Places Norway *Hen, Buskerud, a village in Ringer ...
nbsp; , , Hui Chen (2006), "Construction of a kind of abundant semigroups", ''Mathematical Communications'' (11), 165–171 (Accessed on 25 April 2009) , -valign="top" , elg, , M. Delgado, ''et al.'', ''Numerical semigroups''

(Accessed on 27 April 2009) , -valign="top" , dwa, , P. M. Edwards (1983), "Eventually regular semigroups", ''Bulletin of Australian Mathematical Society'' 28, 23–38 , -valign="top" , ril, , P. A. Grillet (1995). ''Semigroups''. CRC Press. , -valign="top" , ari, , K. S. Harinath (1979), "Some results on ''k''-regular semigroups", ''Indian Journal of Pure and Applied Mathematics'' 10(11), 1422–1431 , -valign="top" , owi, , J. M. Howie (1995), ''Fundamentals of Semigroup Theory'',
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, -valign="top" , agy, , Attila Nagy (2001). ''Special Classes of Semigroups''.
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. , -valign="top" , et, , M. Petrich, N. R. Reilly (1999). ''Completely regular semigroups''.
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. , -valign="top" , hum    , , 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 World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various ...
, (pp. 303–334) , -valign="top" , vm, , ''Proceedings of the International Symposium on Theory of Regular Semigroups and Applications'',
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, 1986 , -valign="top" , ela, , A. V. Kelarev, ''Applications of epigroups to graded ring theory'', Semigroup Forum, Volume 50, Number 1 (1995), 327-350 , -valign="top" , KM, , 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" , igg, , , -valign="top" , in, , , -valign="top" , ennemore, , {{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