MODULOIDS AND PSEUDOMODULES .1. DIMENSION THEORY

The concept of moduloid over a dioid has been introduced in Gondran and Minoux [8] for the algebraic structure left invariant under the action of a matrix A with entries in a dioid (the 'space' of proper 'vectors' of A). Very close structures have also been proposed in the recent years for the study of diverse phenomena which are now identified by the generic name of Discrete Event Dynamical Systems (DEDS). Although various concepts of independence have been proposed, our choice to select a very weak independence property, together with the assumption that the dioid of scalars is completely ordered, perfectly fits the requirements needed for a dimension theory (existence and 'uniqueness' theorems for bases). Moreover, the concept of independence adopted is closely related to the concept of irreducibility in a lattice, and thus shows the links between DEDS's, lattice theory, and classical linear algebra. We also show that, unlike in classical vector spaces, the dimension alone does not characterize the structure. Through various examples, some intuition for complementary investigations on the additional algebraic invariants needed for the classification problem is also provided.