The first book on commutative semigroups was Redei's The theory of .finitely generated commutative semigroups, published in Budapest in 1956. Subsequent years have brought much progress. By 1975 the structure of finite commutative semigroups was fairly well understood. Recent results have perfected this understanding and extended it to finitely generated semigroups. Today's coherent and powerful structure theory is the central subject of the present book. 1. Commutative semigroups are more important than is suggested by the stan dard examples ofsemigroups, which consist ofvarious kinds oftransformations or arise from finite automata, and are usually quite noncommutative. Commutative of factoriza semigroups provide a natural setting and a useful tool for the study tion in rings. Additive subsemigroups of N and Nn have close ties to algebraic geometry. Commutative rings are constructed from commutative semigroups as semigroup algebras or power series rings. These areas are all subjects of active research and together account for about half of all current papers on commutative semi groups. Commutative results also invite generalization to larger classes of semigroups. Archimedean decompositions, a comparatively small part oftoday's arsenal, have been generalized extensively, as shown for instance in the upcoming books by Nagy  and Ciric .
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abelian group addition Algebra archimedean component archimedean semigroup assume C-class called cancellative Chapter cohomology commutative monoid commutative semigroups complete congruence consists constructed contains Conversely Corollary cyclic defined determined direction set element equality equivalence Example exists extent cell family face finite follows free commutative given Grillet group coextension group-free Hence holds homomorphism ideal idempotent identity element implies induced injective integer intersection isomorphism least Lemma mapping Math minimal elements monoid Moreover multiplication natural nilmonoid nilsemigroup nonempty overpath pairs partial particular Ponizovsky family positive presentation projection Proof proper properties Proposition proved provides quotient reduced Rees relation Russian satisfies semilattice smallest strand studied subcomplete subdirect product subdirectly irreducible subgroup subsemigroup subset surjective symmetric Tamura Theorem theory thin trace trivial union uniquely universal yields