mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (just notation, not necessarily th ...

is a

nonempty set In mathematics, the empty set or void 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, whi ...

together with an

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

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, a binary operation ...

. A special class of semigroups is a

class Class, Classes, 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 d ...

s satisfying additional

properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing 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. Perhaps most familiar as a pr ...

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 The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

. 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 a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

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, a 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 a ...

s of the underlying set. 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 Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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

classification Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identif ...

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 iden ...

. 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. 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

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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, HEN or Hens may also refer to: Places Norway *Hen, Buskerud, a village in R ...

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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 technol ...

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Oxford University Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press 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 corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...

. , -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, with more than 170 journals in var ...

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University of Kerala University of Kerala (formerly known as University of Travancore) is a state-run public university in Thiruvananthapuram, the state capital of Kerala, India. It was established in 1937 by a promulgation of the Maharajah of Travancore, Chithi ...

Thiruvananthapuram Thiruvananthapuram ( ), also known as Trivandrum, is the Capital city, capital city of the Indian state of Kerala. As of 2011, the Thiruvananthapuram Municipal Corporation had a population of 957,730 over an area of 214.86 sq. km, making it the ...

India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...

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