mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 other t ...

together with an

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

. 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 differentl ...

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 and ...

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 o ...

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 marked ...

semigroups consists of those semigroups for which the

underlying set In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite se ...

has finite

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

. Members of the class of

Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let '' ...

s 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 a ...

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 s ...

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

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of the

. The underlying sets are not assumed to carry any other mathematical

structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

s like order or

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. As in any algebraic theory, one of the main problems of the theory of semigroups is the

classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...

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 In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topologica ...

. 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 Alfred Hoblitzelle Clifford (July 11, 1908 – December 27, 1992) was an American mathematician born in St. Louis, Missouri who is known for Clifford theory and for his work on semigroups. He did his undergraduate studies at Yale University, Yal ...

G. B. Preston Gordon Bamford Preston (28 April 1925 – 14 April 2015) was an English mathematician best known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from Magdalen College, Oxford. He was born in Workington and broug ...

(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).

. , -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 The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information tec ...

. , -valign="top" ,

ari Ari may refer to: People and fictional characters * Ari (name), a name in various languages, including a list of people and fictional characters * Rabbi Isaac Luria (1534–1572), Jewish rabbinical scholar and mystic known also as Ari * Ari (foot ...

, , 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''.

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

. , -valign="top" , et, , M. Petrich, N. R. Reilly (1999). ''Completely regular semigroups''.

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...

. , -valign="top" ,

hum Hum may refer to: Science * Hum (sound), a sound produced with closed lips, or by insects, or other periodic motion * Mains hum, an electric or electromagnetic phenomenon * The Hum, an acoustic phenomenon * Venous hum, a physiological sensation ...

, , 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 f ...

, (pp. 303–334) , -valign="top" , vm, , ''Proceedings of the International Symposium on Theory of Regular Semigroups and Applications'',

University of Kerala University of Kerala, formerly the University of Travancore, is a state-run public university located in Thiruvananthapuram, the state capital of Kerala, India. It was established in 1937 by a promulgation of the Maharajah of Travancore, Chit ...

Thiruvananthapuram Thiruvananthapuram (; ), also known by its former name Trivandrum (), is the capital of the Indian state of Kerala. It is the most populous city in Kerala with a population of 957,730 as of 2011. The encompassing urban agglomeration populati ...

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

, 1986 , -valign="top" , ela, , A. V. Kelarev, ''Applications of epigroups to graded ring theory'',

Semigroup Forum Semigroup Forum (print , electronic ) is a mathematics research journal published by Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigroup theory. Coverage in the journ ...

, 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 Immunoglobulin G (Ig G) is a type of antibody. Representing approximately 75% of serum antibodies in humans, IgG is the most common type of antibody found in blood circulation. IgG molecules are created and released by plasma B cells. Each IgG ...

, , , -valign="top" , in, , , -valign="top" , ennemore, , {{citation , last = Fennemore , first = Charles , doi = 10.1007/BF02573031 , issue = 1 , journal =

, pages = 172–179 , title = All varieties of bands , volume = 1 , year = 1970 , -valign="top" Algebraic structures Semigroup theory