semigroup
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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.
~24 min read
Article
21 sectionsContents
- 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
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.
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".