# Semigroup

A semigroup is an algebraic structure consisting of a non-empty set together with a single binary operation that is associative, meaning that \( (x * y) * z = x * (y * z) \) for all elements \( x, y, z \) of the set. Formally it is an ordered pair \( (S, *) \), where \( S \) is a non-empty set and \( * : S \times S \to S \) is an associative binary operation on \( S \)<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>. The semigroup is the structure obtained from a magma (a set with any binary operation) by requiring associativity, and it sits below the monoid, which additionally requires an identity element, and the group, which additionally requires inverses<sup>[2](https://cnds.constructor.university/courses/mfcs-2025/mfcs-notes.pdf)</sup><sup> • </sup><sup>[3](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>.

| Key fact | Detail |
|---|---|
| Definition | An ordered pair \( (S, *) \) with \( S \) non-empty and \( * \) an associative binary operation on \( S \)<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup> |
| Place in the hierarchy | Magma + associativity = semigroup; adding an identity gives a monoid; adding inverses gives a group<sup>[2](https://cnds.constructor.university/courses/mfcs-2025/mfcs-notes.pdf)</sup><sup> • </sup><sup>[3](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup> |
| Basic examples | \( \mathbb{R} \) under addition and under multiplication; \( \mathbb{R}^n \) and any vector space under vector addition<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup> |
| Non-examples | \( \mathbb{R} \) under subtraction; \( \mathbb{R}^3 \) under the cross product, which is not associative<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup> |
| Central structural tool | Green's relations \( \mathcal{L}, \mathcal{R}, \mathcal{H}, \mathcal{D}, \mathcal{J} \), which partition a semigroup by the ideals its elements generate<sup>[4](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/semigroups-whose-right-ideals-are-finitely-generated/112D2967022C2988DE83BC18CF48F153)</sup> |
| First structure theorem | The Rees Theorem (1940) classifies 0-simple semigroups with a primitive idempotent as Rees matrix semigroups over a group<sup>[5](https://www.ams.org/bookstore/pspdf/hmath-41-prev.pdf)</sup><sup> • </sup><sup>[6](https://www.ams.org/journals/bull/1970-76-04/S0002-9904-1970-12504-6/S0002-9904-1970-12504-6.pdf)</sup> |
| Applied forms | \( C_0 \)-semigroups of operators for evolution equations; transformation semigroups of finite automata<sup>[7](https://arxiv.org/abs/2402.01572)</sup><sup> • </sup><sup>[8](https://www.herts.ac.uk/research/centres/ccsir/algorithms/semigroups-and-automata)</sup> |

## Definition and immediate consequences

The only axiom beyond closure is associativity. Because the operation is associative, parentheses in a product of three or more elements can be dropped without ambiguity, and a product such as \( abc \) is well defined even though the operation is defined only on pairs<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>. The underlying set must be non-empty; some authors allow the empty semigroup, but the standard definition given in most texts does not<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.

Notation is a practical hazard. It is common to speak of "the semigroup \( S \)" rather than \( (S, *) \), but the same set can carry several different operations, and these give different semigroups<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.

## Examples and non-examples

Nearly all the binary operations met in elementary mathematics are associative, so semigroups are abundant<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>:

- \( \mathbb{R} \) under addition is a semigroup, and \( \mathbb{R} \) under multiplication is a different semigroup on the same set, since the operations differ<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.
- \( \mathbb{R}^n \) under addition is a semigroup, and more generally any vector space is a semigroup under vector addition<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.
- Composition of functions is associative, so functions on a set form a semigroup under composition, the full [transformation semigroup](https://www.edgepedia.org/transformation-semigroup)<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.
- \( \mathbb{R} \) under subtraction is not a semigroup, because subtraction is not associative<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.
- \( \mathbb{R}^3 \) under the cross product \( \times \) is not a semigroup, since the cross product is not associative<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>.

A group is a set with an associative binary operation, an identity element \( e \) satisfying \( e * x = x = x * e \), and an inverse for every element<sup>[3](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>. Every group is therefore a semigroup, but many semigroups are not groups: they may lack an identity, lack inverses, or both. String concatenation over an alphabet is a standard example of a semigroup that has an identity (the empty string) only after it is enlarged to a monoid.

## Relation to groups, monoids, and magmas

The hierarchy of one-operation structures is built by adding axioms. A magma is a set with an arbitrary binary operation; requiring associativity yields a semigroup; requiring an identity element yields a monoid; requiring inverses with respect to that identity yields a group<sup>[2](https://cnds.constructor.university/courses/mfcs-2025/mfcs-notes.pdf)</sup><sup> • </sup><sup>[3](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>. In a structure with an identity, a left inverse and a right inverse of the same element are equal, which is one of the small lemmas that make the group axioms economical<sup>[3](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)</sup>.

Dropping the identity and inverse axioms is not a cosmetic weakening. The generality is what lets one theorem apply simultaneously to addition of real numbers, addition of matrices, multiplication of matrices, and composition of functions<sup>[1](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)</sup>. It also means semigroup theory contains phenomena with no group analog, such as elements that are idempotent (\( x = x^2 \)) or a zero element that absorbs every product.

## History of the theory

The algebraic theory of semigroups developed as a generalization of group theory and out of the multiplicative structure of rings<sup>[9](https://old.maa.org/node/448084)</sup>. Clifford and Preston, the authors of the field's standard two-volume monograph, dated the theory's true beginning to 1928, with a paper of fundamental importance by A. H. Clifford (1908–1992)<sup>[9](https://old.maa.org/node/448084)</sup>. A true theory independent of groups and rings emerged around 1940, when [David Rees](https://www.edgepedia.org/david-rees-welsh-mathematician) (1918–2013) published his paper "On semi-groups", developing a notion of simple semigroup and proving the Rees Theorem, a semigroup analog of the Artin–Wedderburn theorem, also known as the Rees–Sushkevich Theorem because it subsumes an earlier finite-case result of Sushkevich<sup>[5](https://www.ams.org/bookstore/pspdf/hmath-41-prev.pdf)</sup>. Clifford's 1941 theorem characterizing semigroups that are unions of groups followed immediately and, unlike the Rees Theorem, has no analog in group or ring theory<sup>[5](https://www.ams.org/bookstore/pspdf/hmath-41-prev.pdf)</sup>.

The 1940s saw rapid expansion: Paul Dubreil's Paris-based school of "demi-groupes" focused on congruences, and the Soviet school of semigroup theory was reborn<sup>[5](https://www.ams.org/bookstore/pspdf/hmath-41-prev.pdf)</sup>. The field also faced early criticism; G. B. Preston recalled that some mathematicians said semigroups were objects too simple to be interesting and that useful mathematics would not come from studying them<sup>[10](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)</sup>. Clifford and Preston's two-volume *The Algebraic Theory of Semigroups* laid out the basics of the theory, and an international journal devoted to the subject, *Semigroup Forum*, was founded in 1970<sup>[11](https://www.cambridge.org/core/books/abs/semigroup-theory-and-its-applications/foreword/29E674A1B72288E0E5D33EAAFF56249A)</sup><sup> • </sup><sup>[9](https://old.maa.org/node/448084)</sup>.

## Structure theory: Green's relations and the Rees Theorem

To analyze a semigroup's structure, the key instruments are the five [Green's relations](https://www.edgepedia.org/greens-relations) \( \mathcal{L}, \mathcal{R}, \mathcal{H}, \mathcal{D} \), and \( \mathcal{J} \)<sup>[4](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/semigroups-whose-right-ideals-are-finitely-generated/112D2967022C2988DE83BC18CF48F153)</sup>. Two elements are \( \mathcal{L} \)-related if they generate the same principal left ideal, \( \mathcal{R} \) is defined dually with principal right ideals, \( \mathcal{H} \) is the intersection \( \mathcal{L} \cap \mathcal{R} \), \( \mathcal{D} \) is the join of \( \mathcal{L} \) and \( \mathcal{R} \), and \( \mathcal{J} \)-related elements generate the same principal two-sided ideal<sup>[4](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/semigroups-whose-right-ideals-are-finitely-generated/112D2967022C2988DE83BC18CF48F153)</sup><sup> • </sup><sup>[12](https://doi.org/10.1016%2Fj.jsc.2018.01.002)</sup>.

The Rees Theorem, semigroup theory's first major structure theorem, classifies the 0-simple semigroups that have a primitive idempotent. Such a semigroup is isomorphic to a Rees matrix semigroup: take sets \( A \) and \( B \), a group \( G \), and a \( B \times A \) matrix \( C \) with entries in \( G \) or zero, nonzero at least once in each row and column; the elements are \( A \times B \) matrices with at most one nonzero entry, multiplied by \( X \cdot Y = XCY \)<sup>[6](https://www.ams.org/journals/bull/1970-76-04/S0002-9904-1970-12504-6/S0002-9904-1970-12504-6.pdf)</sup>. In the proof, \( A \) indexes the nonzero \( \mathcal{R} \)-classes, \( B \) the nonzero \( \mathcal{L} \)-classes, and \( G \) an \( \mathcal{H} \)-class containing an idempotent<sup>[6](https://www.ams.org/journals/bull/1970-76-04/S0002-9904-1970-12504-6/S0002-9904-1970-12504-6.pdf)</sup>.

## Classes defined by equations

Many important classes of semigroups are captured by quantified equations. Regular semigroups are defined by \( (\forall a)(\exists x) : a = axa \), and completely regular semigroups, in which every element has an inverse with which it commutes, by \( (\forall a)(\exists x) : (a = axa) \wedge (ax = xa) \)<sup>[13](https://link.springer.com/article/10.1007/s00233-023-10397-4)</sup>. Higgins and Jackson proved in 2023 that an elementary class of semigroups closed under homomorphic images of direct products is defined by such equation systems, and that an EHP-class of semigroups is definable by purely existential equations exactly when it is closed under taking containing semigroups, a dual to [Birkhoff's theorem](https://www.edgepedia.org/birkhoffs-theorem)<sup>[13](https://link.springer.com/article/10.1007/s00233-023-10397-4)</sup>.

## Semigroup algebras

A [semigroup algebra](https://www.edgepedia.org/semigroup-algebra) \( \Phi[S] \) over a field \( \Phi \) is the vector space of formal finite sums \( \sum_{s \in S} a_s s \) with basis \( S \), with the semigroup multiplication extended linearly to an algebra structure; when \( S \) is a group this is the group algebra<sup>[14](https://encyclopediaofmath.org/wiki/Semi-group_algebra)</sup>. Describing the linear representations of \( S \) is equivalent to describing the representations of \( \Phi[S] \), which lets the tools of algebra theory be applied to semigroup representations<sup>[14](https://encyclopediaofmath.org/wiki/Semi-group_algebra)</sup>. For a finite [inverse semigroup](https://www.edgepedia.org/inverse-semigroup), an analog of Maschke's theorem holds: \( \Phi[S] \) is semi-simple if and only if the characteristic of \( \Phi \) is zero or a prime that does not divide the order of any subgroup of \( S \)<sup>[14](https://encyclopediaofmath.org/wiki/Semi-group_algebra)</sup><sup> • </sup><sup>[16](https://eprints.whiterose.ac.uk/173601/1/version3.6.pdf)</sup>.

## Applications

**Operator semigroups.** One-parameter semigroups of operators describe processes that change over time and are associated with evolution equations of the form \( x'(t) = Ax(t) \), where \( A \) is a linear, often unbounded, operator on a [Banach space](https://www.edgepedia.org/banach-space). \( C_0 \)-semigroups (strongly continuous semigroups) are generated by partial differential equations and describe the distributions of Markov processes such as diffusion processes, piecewise deterministic Markov processes, and hybrid stochastic processes; stochastic semigroups related to Markov chains, diffusion, jump processes, and stochastic billiards are standard objects of study, with long-time behavior analyzed through tools such as the [Perron–Frobenius theorem](https://www.edgepedia.org/perron-frobenius-theorem) and spectral decomposition<sup>[7](https://arxiv.org/abs/2402.01572)</sup>.

**Automata and computation.** In algebraic automata theory, every finite discrete dynamical system has an associated canonical algebraic structure called a semigroup, and such systems can be decomposed and synthesized hierarchically from irreducible components using Krohn–Rhodes theory, whose prime components are finite simple groups<sup>[8](https://www.herts.ac.uk/research/centres/ccsir/algorithms/semigroups-and-automata)</sup>. The syntactic monoid of a language connects finite automata and regular languages to algebra, and Schützenberger's theorem characterizes star-free languages<sup>[15](https://www.geometry-algebra.polimi.it/en/teaching/phd/semigroup/)</sup>. Algorithmic questions are active: using a variant of Schreier's theorem and Green's relations, computation of an arbitrary subsemigroup of a finite regular semigroup can be reduced to that of associated subgroups, enabling tests of membership and computation of size, Green's structure, idempotents, and maximal subgroups<sup>[12](https://doi.org/10.1016%2Fj.jsc.2018.01.002)</sup>.


## References

1. [Chapter 4: Semigroups, MATH2301 course notes, University of Queensland](https://courses.smp.uq.edu.au/MATH2301/Ch4.pdf)
2. [Mathematical Foundations: Abstract Algebra, Jürgen Schönwälder, Constructor University](https://cnds.constructor.university/courses/mfcs-2025/mfcs-notes.pdf)
3. [Abstract Algebra, Chapter 2: Group theory, Romyar Sharifi, UCLA](https://math.ucla.edu/~sharifi/notes/algebra-ch02.html)
4. [Semigroups whose right ideals are finitely generated, Proceedings of the Edinburgh Mathematical Society](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/semigroups-whose-right-ideals-are-finitely-generated/112D2967022C2988DE83BC18CF48F153)
5. [Algebra at the Beginning (preview), American Mathematical Society](https://www.ams.org/bookstore/pspdf/hmath-41-prev.pdf)
6. [Book review: The algebraic theory of semigroups, Vol. I and II, by A. H. Clifford and G. B. Preston, Bulletin of the AMS (1970)](https://www.ams.org/journals/bull/1970-76-04/S0002-9904-1970-12504-6/S0002-9904-1970-12504-6.pdf)
7. [Transformation semigroups and their applications, Katarzyna Pichór and Ryszard Rudnicki, arXiv:2402.01572](https://arxiv.org/abs/2402.01572)
8. [Semigroups and automata, University of Hertfordshire Algorithms Research Group](https://www.herts.ac.uk/research/centres/ccsir/algorithms/semigroups-and-automata)
9. [Mathematics Across the Iron Curtain: A History of the Algebraic Theory of Semigroups, Mathematical Association of America](https://old.maa.org/node/448084)
10. [Gordon Preston on semigroups, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)
11. [Foreword, Semigroup Theory and its Applications, Cambridge University Press](https://www.cambridge.org/core/books/abs/semigroup-theory-and-its-applications/foreword/29E674A1B72288E0E5D33EAAFF56249A)
12. [Computing finite semigroups, Journal of Symbolic Computation](https://doi.org/10.1016%2Fj.jsc.2018.01.002)
13. [Equationally defined classes of semigroups, Peter M. Higgins and Marcel Jackson, Semigroup Forum 107, 459–477 (2023)](https://link.springer.com/article/10.1007/s00233-023-10397-4)
14. [Semi-group algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Semi-group_algebra)
15. [(Semi)group theory and automata, Politecnico di Milano PhD course](https://www.geometry-algebra.polimi.it/en/teaching/phd/semigroup/)
16. [eprints.whiterose.ac.uk](https://eprints.whiterose.ac.uk/173601/1/version3.6.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Abstract algebra — overview*

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