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On divisor-closed submonoids and minimal distances in finitely generated monoids. (English) Zbl 07040044
For a finitely generated commutative cancellative monoid \(H\), the authors study the set \(\Delta^*(H)\) of minimal distances occurring in the theory of non-unique factorizations, developed in the book by A. Geroldinger and F. Halter-Koch [Non-unique factorizations. Algebraic, combinatorial and analytic theory. Boca Raton, FL: Chapman & Hall/CRC (2006; Zbl 1113.11002)]. They show that the divisor-closed submonoids \(A\) of \(H\) form a finite lattice (Theorem 4), determine the sets of generators for them, and show how to compute \(\Delta^*(A)\). In the case when \(H\) is an affine semigroup a geometric approach is used to describe its divisor-closed submonoids (Theorem 15). This is used to present an algorithm to compute \(\Delta^*(H)\) for every finitely generated \(H\).
MSC:
13A05 Divisibility and factorizations in commutative rings
20M13 Arithmetic theory of semigroups
11R27 Units and factorization
20M32 Algebraic monoids
68W30 Symbolic computation and algebraic computation
20M14 Commutative semigroups
52B11 \(n\)-dimensional polytopes
Software:
Normaliz; Python
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References:
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