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].
