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 [2001] and Ciric
[2002].
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