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Abstract

The cross number $\kappa (a)$ can be defined for any element $a$ of a Krull monoid. The property $\kappa (a) = 1$ is important in the study of algebraic numbers with factorizations of distinct lengths. The arithmetic meaning of the weaker property, $\kappa (a) \in {\mathbb Z}$, is still unknown, but it does define a semigroup which may be interesting in its own right. This paper studies some arithmetic (divisor theory) and analytic (distribution of elements with a given norm) properties of that semigroup and a related semigroup of ideals.