On ideals of cone preserving maps

Abstract

Let C be a full cone in a finite-dimensional real vector space V, and let Hom( C) denote the semiring of linear maps A of V such that A( C) ⊂ C. A given species of ideal in Hom( C) is defined as would be expected from the similarity between rings and semirings. The minimal one-sided ideals of Hom( C) are identified, assuming C is polyhedral, and using these it is shown that if C 1, C 2 are two such cones such that Hom( C 1) and Hom( C 2) are isomorphic as semirings then C 1 and C 2 are linearly isomorphic. In contrast, maximal one-sided ideals will not play such a role in general. For if C is an irreducible cone, Hom( C) has a unique maximal ideal, and conversely. Nevertheless, the maximal one-sided ideals of the semiring N n of n × n non-negative matrices are identified. Finally, it is shown that for an arbitrary full cone C, Hom( C) never satisfies the descending-chain condition on principal one-sided ideals (if dim V >1).