Groupe d’études et de recherche en analyse des décisions

# The Geometry of Finite Dimensional Pseudomodules

## Edouard Wagneur

A semimodule M over an idempotent semiring P is also idempotent. When P is linearly ordered and conditionnally complete, we call it a pseudoring, and we say that M is a pseudomodule (p.m.) over P. The classification problem of the isomorphy classes of p.m.'s is a combinatorial problem which, in part, is related to the classificationof isomorphy classes of semilattices. We define the structural semilattice of a p.m., which is then used to introduce the concept of torsion. Then we show that every finitely generated p.m. may be canonically decomposed into the "sum" of a torsion free sub-p.m., 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 a sub-pseudomodule of the free p.m. over n generators, the torsion free part, also called semiboolean, is completely charecterized 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 Pn with n generators will also be presented.

, 19 pages