Non-semigroup gradings of associative algebras

TL;DR

This paper demonstrates that associative algebras can have non-semigroup gradings, extending the known phenomenon from Lie algebras to associative algebra structures.

Contribution

The paper provides the first example of non-semigroup gradings within associative algebras, showing a new class of algebraic gradings.

Findings

Existence of associative algebras with non-semigroup gradings

Extension of non-semigroup grading concept from Lie to associative algebras

New examples illustrating non-associative binary operations on grading sets

Abstract

It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.