Monoids determined by a homogenous linear diophantine equation and the half-factorial property

Abstract

We study additive submonoids M of N n consisting of the solutions of a homogeneous linear diophantine equation with integer coefficients. Surprisingly, not very much is known about the structure of M . M is a Krull monoid which, however, cannot be realized as a multiplicative monoid of a Krull domain. The concepts of divisor theory and divisor class group, nevertheless, do apply and we use them to characterize the factoriality of M in terms of the coefficients of the diophantine equation. In this paper, we concentrate on the more difficult question of finding conditions under which M is half-factorial. Since the famous Carlitz criterion of class number at most two breaks down for the Krull monoid M , we develop some new sufficient and/or necessary conditions for the half-factoriality of M . Among others, we present a geometric criterion for M to be half-factorial and an inequality condition on the coefficients of the diophantine equation assuring the half-factoriality of M .