Stratified Algebra

TL;DR

This paper introduces Stratified Algebra, a new layered algebraic framework that models systems with local consistency and global asymmetry, expanding traditional algebraic structures with context-dependent interactions.

Contribution

It formalizes the concept of Stratified Algebra, providing axioms, models, and analysis of layered algebraic systems with diverse interaction patterns.

Findings

Defined axioms for stratified interactions

Developed matrix-based models satisfying stratification axioms

Explored implications for algebraic dynamics and symmetry

Abstract

We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their implications for algebraic dynamics, and present concrete matrix-based models that satisfy different subsets of these axioms. Both associative and bracket-sensitive constructions are considered, with an emphasis on stratum-breaking propagation and permutation symmetry. This framework proposes a paradigm shift in the way algebraic structures are conceived: instead of enforcing uniform global rules, it introduces stratified layers with context-dependent interactions. Such a rethinking of algebraic organization allows for the modeling of systems where local consistency coexists with global asymmetry, non-associativity, and semantic transitions.