AbstractA 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 isomorphism classes of pseudomodules is a combinatorial problem which, in part, is related to the classification of isomorphism 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 sub-pseudomodule, 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 a great simplification of the classification problem, since each part can be studied separately. For subpseudomodules of the free pseudomodule over m generators, we conjecture that 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 isomorphism class of a torsion sub-pseudomodule of Pm with m generators will also be presented.

Pseudorings and Pseudomodules

A pseudoring (P, +, ·) is an idempotent, commutative, completely ordered, and conditionally complete semiring with minimal element 0, such that (P\ {0}, ·) is an integrally closed commutative group. For simplicity, the reader may think of P as R∪ {–∞} (or R∪ {∞}) with + the max (resp. the min) operator, and o the usual addition.