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Linear Dependence and Independence in Semi-Modules and Moduloids

Part of the Operations Research/Computer Science Interfaces book series (ORCS, volume 41)

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.

Keywords

Linear Dependence Assignment Problem Chromatic Number Dependence Relation Complete Bipartite Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2008

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