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Moduloïds and pseudomodules. I: Dimension theory. (English) Zbl 0757.06008
The author selects the concept of a moduloïd over a dioïd to have a very weak independence property, together with the assumption that the dioïd of scalars is completely ordered, that perfectly fits the requirements needed for a dimension theory. Moreover, the concept of independence adopted is closely related to the concept of irreducibility in a lattice, and thus shows the links between Discrete Event Dynamical Systems, lattice theory, and classical linear algebra. The author also shows that, unlike in classical vector spaces, the dimension alone does not characterize the structure. Through various examples, some intuition for complementary investigations on the additional algebraic invariants needed for the classification problems is also provided.
In Section 2. “Basic Definitions”, the author first gives a motivating example, and then states the basic definitions and elementary properties which will be needed. Some examples are also provided. In Section 3. “Moduloïds and weak bases”, the author states and proves the fundamental theorems related to bases and dimension for moduloïds. The corresponding results for pseudomodules are given in Section 4. “The case of pseudomodules. Bases and dimension”. Some additional remarks and examples then conclude the paper in Section 5. “Concluding remarks and open problems”.
Reviewer: Y.Kuo (Knoxville)

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F25 Ordered rings, algebras, modules
15A99 Basic linear algebra
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