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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
Normaliz; Python
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