Membership deformation of commutativity and obscure $n$-ary algebras

TL;DR

This paper introduces a novel approach to deforming algebraic structures by incorporating fuzzy set membership functions into commutation relations, extending to n-ary algebras and their representations.

Contribution

It proposes a new mechanism for deforming algebraic commutativity using fuzzy set memberships, extending to n-ary algebras and their projective representations.

Findings

Defined deformation of commutativity via fuzzy membership functions

Extended deformation concepts to n-ary algebras and their representations

Provided a framework for deforming $ ext{ε}$-Lie and Weyl algebras

Abstract

A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and $\varepsilon $-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" $\varepsilon$-Lie algebras and Weyl algebras. Further, the above constructions are extended to $n$-ary algebras for which the projective representations and $\varepsilon $-commutativity are studied.