semigroup

Also known as associative magma

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. In mathematical analysis, the term also appears in the theory of one-parameter operator semigroups: see C0-semigroup.

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Vinony's link graph records 307 inbound references to semigroup, and connects out to identity element, magma and quasigroup.

Vinony files it under Algebraic structures and Semigroup theory.

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Wikidata facts

Has part: binary operation

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Stack Exchange tag: stackoverflow.com/tags/semigroup
has characteristic: closure
maintained by WikiProject: WikiProject Mathematics

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Encyclopedic overview

21 sections

Contents

Algebraic overview
Definition
Examples of semigroups
Basic concepts
Identity and zero
Subsemigroups and ideals
Homomorphisms and congruences
Quotients and divisions
Structure of semigroups
Special classes of semigroups
Structure theorem for commutative semigroups
Group of fractions
Semigroup methods in partial differential equations
History
Generalizations
See also
Notes
Citations
References
General references
Specific references

The binary operation of a semigroup is most often denoted multiplicatively: x\cdot y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x,y). Associativity is formally expressed as that (x\cdot y)\cdot z=x\cdot (y\cdot z) for all x, y and z in the semigroup. An example of a semigroup is that formed by string concatenation, which glues together strings. For example, the concatenation of the strings "spot " and "run" is the string "spot " • "run" = "spot run". Associativity means that "See " • ("spot " • "run") = "See " • "spot run" = "See spot run" = "See spot " • "run" = ("See " • "spot ") • "run".

Excerpted from Wikipedia’s “semigroup” article, available under the CC BY-SA 4.0 licence.