Arithmetical semigroups. I: Direct factors
The authors generalize results of Daboussi, Erdős, Saffari and Vaughan on direct factors of the set \({\mathbb{N}}\) of positive integers to arithmetical semigroups (G,\(\partial)\) with a subset \(P\subset G\) of free generators (``primes'') and with an identity e, where the map \(\partial\) satisfies \(\partial (e)=0\), \(\partial (g)>0\), if \(g\neq e\), \(\partial (a\cdot b)=\partial (a)+\partial (b)\), and \(\gamma (n):=\#\{g\in G; \partial (g)=n\}\) is finite. Write \(G=A\otimes B\), if every g in G may be written uniquely as a product \(a\cdot b\), \(a\in A\), \(b\in B\). Then the authors prove: (1) Assume \(\gamma (n)\sim c\cdot q^ n\) for \(n\to \infty\), where \(q>1\), \(c>0\). If \(G=A\otimes B\), then \[ \lim_{n\to \infty}(\#\{a\in A; \partial (a)=n\}/\gamma (n))=(1/\sum_{b\in B}q^{-\partial (b)}), \] where the right-hand-side is zero, if \(\sum_{b\in B}q^{-\partial (b)}\) diverges. (2) Assume \(\gamma (n)=c\cdot q^ n+O(q^{\nu n})\), where \(q>1\), \(0\leq \nu <1\), \(c>0\). Let \(G=A\otimes B\); if \(\sum_{p\in A\cap P}q^{- \partial (p)}\) converges, then \(\sum_{a\in A}q^{-\partial (a)}\) converges too, and some stronger result holds, if \(A\cap P=\emptyset.\) Some open problems are mentioned and partial solutions for these are given.
direct factors of commutative additive arithmetical semigroups
direct factors of the set \({\mathbb{N}}\) of positive integers
scientific article; zbMATH DE number 4191809