Translations of commutative unique factorization semigroups

Abstract

Every commutative semigroup is a semilattice of archimedean subsemigroups. Given a semilattice Y and a collection of commutative semigroups S indexed by Y, the converse problem of constructing every commutative semigroup which is a semilattice (Y) of the semigroups S remains unsolved except in certain cases. If each S is an abelian group the result follows as a special case of Clifford's Theorem (see [33, Thm. 4.11). The authors [13 have given a solution for any semilattice where each S is assumed to be cyclic, and more recently, Tamura has solved the problem for exclusive semilattices of exclusive semigroups [63 For a characterization of the general semilattice decomposition when each Se is weakly reductive, see Theorem 7.8.13 of [41. An element x of a commutative semigroup S is said to be prime if x ~ S 2 . S is said to have unique factorization if each nonzero element of S can be written uniquely as a product of primes in the usual sense. A commutative semigroup has unique factorization if and only if it is free or a Rees quotient of a free commutative semigroup (E2~, Thm. i). Commutative nil unique factorization semigroups were used in the characterization of arbitrary commutative nil semigroups in [11 (see [53 for an alternate characterization). Since commutative unique factorization semigroups arise naturally in the general theory of commutative semigroups, it seems of interest