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a fundamental concept of modern algebra. A semigroup is a set on which an associative operation is defined. The concept is a generalization of the concept of a group whereby only one of the group axioms remains; hence the term “semigroup.”
Examples of semigroups are very numerous in mathematics and include various sets of numbers with the operation of addition or multiplication, provided the sets are closed under the operation (that is, they contain the sum or product, respectively, of any two elements), the semigroup of matrices under multiplication, the semigroup of functions under multiplication, and the semigroup of sets under the operation of intersection or union. One of the simplest examples of a semigroup is the set of all natural numbers under the operation of addition; this semigroup is a part, or subsemigroup, of the group of integers under addition—that is, it can be embedded in the group of integers. Not every semigroup, however, can be embedded in a group.
The following example of a semigroup is important in the general theory and in certain applications. Let X be an arbitrary set, and let the operation * be defined on the set Fx of all finite sequences of elements of X by means of the formula
(x1 …, xn) * (y1, …, ym)
= (x1,…, xn, y1,…, ym)
Fx is a semigroup under the operation * and is called a free semigroup on X. Every semigroup is the homomorphic image of some free semigroup.
If a set of transformations of an arbitrary set M is closed under the operation of composition, it is a semigroup with respect to this operation. Such a semigroup, for example, is the set of all transformations of M, which is called a symmetric semigroup on M. Many important sets of transformations are semigroups, although often they are not groups. On the other hand, every semigroup is isomorphic to some semigroup of transformations. Thus, the semigroup is the most suitable concept for the study of transformations in their most general form. It is largely through the consideration of transformations that links are found between the theory of semigroups and other areas of mathematics, such as modern differential geometry, functional analysis, and the abstract algebraic theory of automata.
The first investigations devoted to semigroups took place in the 1920’s. By the end of the 1950’s, the theory of semigroups was an independent branch of modern algebra that continued to undergo active development. The study of the abstract (that is, not dependent on the specific nature of the elements) properties of various associative operations is the subject of algebraic semigroup theory, one of whose main tasks is the description of the structure of different semigroups and the classification of semigroups. By imposing additional restrictions on the semigroup operation, we can define a number of important types of semigroups, such as completely simple semigroups and inverse semigroups. An important part of the general theory is the theory of the representation of semigroups by transformations and matrices. The introduction into semigroups of additional structures consistent with the semigroup operation distinguishes special branches of the theory of semigroups, such as the theory of topological semigroups.
REFERENCESSushkevich, A. K. Teoriia obobshchennykh grupp. Kharkov-Kiev, 1937.
Liapin, E. S. Polugruppy. Moscow, 1960.
Clifford, A. H., and G. B. Preston. Algebraicheskaia teoriia polugrupp, vols. 1–2. Moscow, 1972. (Translated from English.)
Hofmann, K., and P. Mostert. Elements of Compact Semigroups. Columbus, Ohio, 1966.
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