Stratified Algebra
Stanislav Semenov
TL;DR
This work introduces Stratified Algebra, a framework where a finite-dimensional vector space $V$ is partitioned into disjoint strata $V_\alpha$ and equipped with a binary operation $*$ that acts differently within and across strata. The authors formulate four axioms (SA1)–(SA4) governing intra-stratum associativity, cross-layer asymmetry, layered permutation symmetry, and bracket-sensitive transitions, and provide concrete matrix-based and nonlinear realizations that satisfy subsets or all four axioms. They analyze linear, higher-dimensional, nonlinear, and finite-field instantiations to demonstrate how local coherence within strata coexists with global nonuniformity, and how bracket patterns can drive semantic transitions across layers. The work offers constructive models, symbolic verifications, and suggests potential applications in symbolic computation, hierarchical semantics, and cryptography, laying groundwork for a systematic theory of stratified, context-driven algebraic systems. Overall, the paper establishes the existence and practicality of fully stratified algebras and outlines directions for classification, morphisms, and algorithmic stratification analysis, highlighting a novel algebraic paradigm with rich internal dynamics.
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.