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 is a nonempty set 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 of semigroups satisfying additional 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 semigroups consists of those semigroups for which the underlying 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 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

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 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: * 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 d ...

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 (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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elg Elg or ELG may refer to: People * Jarno Elg (born 1975), Finnish convict * Taina Elg (born 1930), Finnish-American actress and dancer * Øivind Elgenes (born 1958), Norwegian singer Transportation * El Golea Airport, in Algeria * Elgin railw ...

, , 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 RIL may refer to: * Radio Interface Layer, a software interface used in a mobile device to communicate via mobile networks * RDF Inference Language, a means of expressing expert systems rules and queries that operate on RDF models * Reliance Indu ...

, , 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 ...

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