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 other ...

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 ...

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 differently ...

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 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 Traditionally, a finite verb (from la, fīnītus, past particip ...

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 set ...

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 ...

ic 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

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

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

or topology. 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. 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

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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 brough ...

(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, ...

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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 techn ...

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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 (fo ...

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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, Ch ...

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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 jou ...

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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 an ...

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

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