Editing Semigroup (section)

== History ==
The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as [[group (mathematics)|groups]] or [[ring (algebra)|rings]]. A number of sources<ref>{{cite web| url = http://jeff560.tripod.com/s.html| title = Earliest Known Uses of Some of the Words of Mathematics}}</ref><ref name=Hollings>{{cite web| url = http://uk.geocities.com/cdhollings/suschkewitsch3.pdf| archive-url = https://www.webcitation.org/5kmeaTTMB?url=http://uk.geocities.com/cdhollings/suschkewitsch3.pdf| url-status = dead| archive-date = 2009-10-25| title = An account of Suschkewitsch's paper by Christopher Hollings}}</ref> attribute the first use of the term (in French) to J.-A. de Séguier in ''Élements de la Théorie des Groupes Abstraits'' (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's ''Theory of Groups of Finite Order''.

[[Anton Sushkevich]] obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finite [[simple semigroup]]s and showed that the minimal ideal (or [[Green's relations]] J-class) of a finite semigroup is simple.<ref name=Hollings/> From that point on, the foundations of semigroup theory were further laid by [[David Rees (mathematician)|David Rees]], [[James Alexander Green]], {{ill|Evgenii Sergeevich Lyapin|fr|Evgueni Liapine}}, [[Alfred H. Clifford]] and [[Gordon Preston]]. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical called ''[[Semigroup Forum]]'' (currently published by [[Springer Verlag]]) became one of the few mathematical journals devoted entirely to semigroup theory.

The [[representation theory]] of semigroups was developed in 1963 by [[Boris Schein]] using [[binary relation]]s on a set ''A'' and [[composition of relations]] for the semigroup product.<ref>B. M. Schein (1963) "Representations of semigroups by means of binary relations" (Russian), [[Matematicheskii Sbornik]] 60: 292–303 {{mr|id=0153760}}</ref>  At an algebraic conference in 1972 Schein surveyed the literature on B<sub>''A''</sub>, the semigroup of relations on ''A''.<ref>B. M. Schein (1972) ''Miniconference on semigroup Theory'', {{mr|id=0401970}}</ref> In 1997 Schein and [[Ralph McKenzie]] proved that every semigroup is isomorphic to a transitive semigroup of binary relations.<ref>B. M. Schein & R. McKenzie (1997) "Every semigroup is isomorphic to a transitive semigroup of binary relations", [[Transactions of the American Mathematical Society]] 349(1): 271–85 {{mr|id=1370647}}</ref>

In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like [[inverse semigroup]]s, as well as monographs focusing on applications in [[algebraic automata theory]], particularly for finite automata, and also in [[functional analysis]].