Multiary gradings

TL;DR

This paper develops a comprehensive theory of multiary graded polyadic algebras, extending classical group-graded algebras to higher-arity structures, revealing new phenomena and classification results.

Contribution

It introduces multiary group gradings, explores compatibility conditions, and establishes foundational theorems for polyadic algebras, expanding the algebraic framework beyond binary cases.

Findings

Introduction of multiary group gradings

Classification of graded homomorphisms

Existence of higher power gradings and arity constraints

Abstract

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, the First Isomorphism Theorem for graded polyadic algebras and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility.