The geometry of finite dimensional pseudomodules

A semimodule M over an idempotent semiring P is also idempotent. When P is linearly ordered and conditionally complete, we call it a pseudoring, and we say that M is a pseudomodule over P. The classification problem of the isomorphy classes of pseudomodules is a combinatorial problem which, in part, is related to the classification of isomorphy classes of semilattices. We define the structural semilattice of a pseudomodule, which is then used to introduce the concept of torsion. Then we show that every finitely generated pseudomodule may be canonically decomposed into the “sum” of a torsion free subpseudomodule, and another one which contains all the elements responsible for the torsion of M. This decomposition is similar to the classical decomposition of a module over an integral domain into a free part, and a torsion part. It allows for a great simplification of the classification problem, since each part can be studied separately. In sub-pseudomodules of the free pseudomodule over m generators, the torsion free part, also called semiboolean, is completely characterized by a weighted oriented graph whose set of vertices is the structural semilattice of M. Partial results on the classification of the isomorphy class of a torsion sub-pseudomodule of P^m with m generators will also be presented