Groupe d’études et de recherche en analyse des décisions


Direct and Semi-direct Sum Decomposition of Idempotent Semimodules

A module over a principal ideal domain splits into a direct sum of a free module and a torsion module. This decomposition does not hold in general for semimodules over a semiring. We give here a necessary and sufficient condition for an idempotent semimodule to be the direct sum of two subsemimodules. We recall the concept of semidirect sum of idempotent semimodules, and then use the concept of torsion and semi-Boolean semimodules introduced in (Wagneur 1991b), 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 clearer view on the existence of a left inverse to the inclusion map for a subsemimodule, stated in (Wagneur 1996a,b) for the direct sum of semimodules. Also, our decomposition suggests how, when used in connexion with the results of (Prou 1997, Prou and Wagneur 1997), we can simplify the classification of finite dimensional semimodules.

, 15 pages