Groupe d’études et de recherche en analyse des décisions

# Dequantisation: Direct and Semi-Direct Sums of Idempotent Semimodules

## Edouard Wagneur

We investigate here the "anatomy" of idempotent semimodules, i.e. we look for the equivalent of the classical decomposition of a module over a principal ideal domain. This is an important step towards the classification of these algebraic structures. The direct sum decomposition does not hold in general for idempotent semimodules over a semifield. We give here a necessary and sufficient condition for such a decomposition. Then we define the slightly more general concept of semi-direct sum, which is more adapted to ordered structures. Similar to the classical theory, where the decomposition of modules is derived from that of abelian groups, we first deal with idempotent abelian monoïds, or, equivalently, with semilattices. We introduce the semi-direct sum of semilattices, extend it to idempotent semimodules, and then introduce the concepts of torsion, Boolean, and semi-Boolean semimodules, to show how a general semimodule over an idempotent semiring splits into a semi-direct sum of a free, a Boolean, a semi-Boolean and a torsion semimodule. This also provides a constructive insight into the existence of a left inverse to the inclusion map of a subsemimodule.

, 18 pages