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Realization theorems for semigroups with divisor theory. (English) Zbl 0751.20045
A commutative semigroup \(S\) with cancellation and unit element is said to have a divisor theory if there exists a homomorphism \(f\) of \(S\) into a free commutative semigroup \(D\) such that every element of \(D\) can be written as the GCD of some elements of \(f(S)\) and for all \(a,b\in S\), if \(f(a)\) divides \(f(b)\) in \(D\) then \(a\) divides \(b\) in \(S\). [L. Skula, Math. Z. 114, 113-120 (1970; Zbl 0177.032)]. The authors show that every such semigroup can be written in the form \(S\cong S^ \times\times B(A)\), where \(S^ \times\) is the group of invertible elements of \(S\), \(A\) is a subset of an Abelian group \(G\) and \(B(A)\) (the block semigroup of \(A\)) consists of all finite sequences of elements of \(A\) with vanishing sum. Moreover the authors introduce arithmetically closed subsemigroups of a semigroup with divisor theory and show that every semigroup with divisor theory is isomorphic (up to units) to an arithmetically closed subsemigroup of the multiplicative semigroup of a Dedekind domain.

MSC:
20M14 Commutative semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
20M10 General structure theory for semigroups
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