Skip to main content


semigroup A very simple algebraic structure comprising a set S on which there is defined an associative operation denoted by ◦ (compare group). The operator ◦ is assumed to take operands from the set and produce results that are also in S. When the set S is finite a semigroup can be described by giving the Cayley table of the operation ◦; otherwise it can be described by giving a rule for ◦.

Examples of semigroups include: strings with the operation of concatenation (joining together); the set of n×n matrices together with the operation of multiplication; the set of transformations of a set and the operation of composing functions; the integers and the operation of choosing the maximum (or minimum) of two elements. The set of integers together with subtraction does not constitute a semigroup.

Semigroups play a major role in the theory of sequential machines and formal languages. If M is a sequential machine then any input string induces a function over the state-set of M. The set of all such induced functions forms a semigroup of the machine under function composition (see Myhill equivalence, Nerode equivalence). Semigroups are also used in certain aspects of computer arithmetic. See also free semigroup, transformation semigroup, monoid.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"semigroup." A Dictionary of Computing. . 22 Feb. 2019 <>.

"semigroup." A Dictionary of Computing. . (February 22, 2019).

"semigroup." A Dictionary of Computing. . Retrieved February 22, 2019 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.