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Irreducible factorization lengths and the elasticity problem within \(\mathbb N\). (English) Zbl 1280.20063

Let \(M\) be a submonoid of \((\mathbb N,\cdot)\) (where \(\mathbb N\) stands for the set of nonnegative integers). An irreducible element or atom in \(M\) is an element that cannot be factored as a product of two nonunits of \(M\). The monoid \(M\) is atomic, that is, every \(x\) can be expressed as a finite product of irreducible elements. Each of these expressions is called a factorization, and the length of a factorization is the number of irreducibles (counting multiplicities) involved in the expression. The elasticity of an element \(x\in M\) is the quotient of the maximal length of a factorization by the minimal possible length. For the whole monoid \(M\), its elasticity is just the supremum of the elasticities of its elements.
A positive integer \(x\) respects \(M\) if for every positive integer \(y\), \(xy\in M\) if and only if \(x^2y\in M\). The monoid \(M\) is said to be R-respectful if there exists a positive integer \(r\) such that \(x^r\) respects \(M\) for all positive integers \(x\).
The radical of \(M\) is defined as the set of primes \(p\) such that \(p^k\in M\) for some positive integer \(k\). The monoid \(M\) is \(t\)-modest, with \(t\) a positive integer, if for every irreducible \(x\in M\), the number of primes in the radical of \(M\) dividing \(x\) does not exceed \(t\). If such a \(t\) exists, then \(M\) is said to be T-modest.
Examples of submonoids of \((\mathbb N,\cdot)\) that are R-respectful and T-modest are congruence monoids (monoids of the form \(\{x\in\mathbb N\setminus\{0\}\mid x\bmod n\in\Gamma\}\cup\{1\}\), with \(n\) a positive integer and \(\Gamma\) a subsemigroup of \((\mathbb Z_n,\cdot)\)), monoids of the form \(\{2^s\mid s\in S\}\), with \(S\) a numerical semigroup, and of the form \(\{2^s3^t\mid s\in S,\;t\in T\}\) with \(S\) and \(T\) numerical semigroups.
The authors prove that if \(M\) is R-respectful and \(t\)-modest for some positive integer \(t\), then the elasticity of \(M\) is upper bounded by \(t\). They also provide a characterization of congruence monoids having finite elasticity.

MSC:

20M13 Arithmetic theory of semigroups
11A51 Factorization; primality
20M14 Commutative semigroups
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