Reducing the axioms of hypergroups, hyperfields, hypermomules and related structures. A new axiomatic basis for hypercompositional structures

TL;DR

This paper refines the axiomatic foundations of hypercompositional algebra structures by identifying dependent axioms and proposing a minimal, independent set of axioms for these structures.

Contribution

It introduces new, minimal axiomatic definitions for hypercompositional structures, improving the foundational understanding of hyperalgebraic systems.

Findings

Axioms in hypercompositional structures are not independent

New minimal axiomatic definitions are proposed

Enhanced clarity in the foundational basis of hyperalgebra

Abstract

This paper is concerned with the axiomatic basis of structures within Hypercompositional Algebra. It is proven that the axioms employed in the definition of numerous hypercompositional structures lack independence. Accordingly, novel definitions are introduced in this work which minimize the established definitions by reducing the necessary set of axioms.