Linear Dependence and Independence in Semi-Modules and Moduloids

The present chapter is devoted to problems of linear dependence and independence in semi-modules (and moduloids). The semi-module structure (resp. the moduloid structure) is the one which arises naturally in the properties of sets of vectors with entries in a semiring (resp. in a dioid). Thus, they turn out to be analogues for algebraic structures on semirings and dioids to the concept of a module for rings.

Section 2 introduces the main basic notions such as morphisms of semi-modules, definitions of linear dependence and independence, generating families and bases in semi-modules. As opposed to the classical case, it will be shown that, in many cases, when a semi-module has a basis, it is unique.

Section 3 is then devoted to studying the links between the bideterminant of a matrix and the concepts of linear dependence and independence previously introduced. Several classical results of linear algebra over vector fields are generalized here to semi-modules and moduloids, in particular those related to selective-invertible dioids and MAX-MIN dioids.