Groupe d’études et de recherche en analyse des décisions

G-99-51

Direct and Semi-Direct Sum Decomposition of Idempotent Semimodules and Semilattices

In classical module theory, a module over a principal ideal domain may be split into the direct sum of a free module and a torsion module. This decomposition does not hold in general for a semimodule over a semiring. We give here a necessary and sufficient condition for an idempotent semimodule to be the direct sum of two subsemimodules. We also introduce the semidirect sum of semilattices, extend it to idempotent semimodules, and then use the concept of torsion and semi-Boolean semimodules introduced in (Wagneur, 1991), to show how a general semimodule over an idempotent semiring may split into a semidirect sum of a semi-Boolean semimodule and a (pure) torsion semimodule. This provides a constructive insight into the existence of a left inverse to the inclusion map of a subsemimodule. Also, our decomposition suggests how the classification of finite dimensional Rmax semimodules may be simplified. As a result, we also get an insight into the classification of idempotent abelian monoids.

, 20 pages