An alternative to modules

TL;DR

This paper introduces a new class of geometrically characterized algebras that extend modules, providing examples and clarifying their properties beyond traditional module theory.

Contribution

It generalizes the concept of modules using geometric properties, expanding the class to include new algebraic structures with coordinate representations.

Findings

Contains all commutative based universal algebras

Provides four new examples of these algebras

Shows these structures are not modules but still have coordinate representations

Abstract

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces that generates these fields. This property also concerned the semi-linear transformations, which are necessary to define geometrical invariance.Yet, the class of such geometrical generalizations was practically unknown. We only knew that it differs from the class of modules, because of a simple example (the sum monoid of natural numbers). Here, we partly clarify the extent of this class: we prove that it contains at least the one of all commutative based universal algebras and we provide it with four new examples. Again, our further examples are not modules. They exhibit a wider choice of both types and algebraic properties, though they keep the representations by "coordinates".