Information entropy re-defined in a category theory context using preradicals

TL;DR

This paper redefines algebraic entropy within a category theory framework using preradicals, establishing a formal, lattice-respecting definition applicable to modules and their associated lattices.

Contribution

It introduces a novel entropy concept for preradicals in module categories and demonstrates its equivalence for lattice preradicals, bridging module theory and lattice theory.

Findings

Entropy for preradicals respects their lattice order

Defined entropy for lattice preradicals is functorially equivalent to module preradicals

Provides a formal, categorical approach to algebraic entropy

Abstract

Algebraically, entropy can be defined for abelian groups and their endomorphisms, and was latter extended to consider objects in a Flow category derived from abelian categories, such as $R\textit{-}Mod$ with $R$ a ring. Preradicals are endofunctors which can be realized as compatible choice assignments in the category where they are defined. Here we present a formal definition of entropy for preradicals on $R$-Mod and show that the concept of entropy for preradicals respects their order as a big lattice. Also, due to the connection between modules and complete bounded modular lattices, we provide a definition of entropy for lattice preradicals, and show that this notion is equivalent, from a functorial perspective, to the one defined for module preradicals.