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arXiv · 2006.07865

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

Steven Duplij
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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.

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Steven Duplij. 2020-06-14. Membership deformation of commutativity and obscure $n$-ary algebras. https://doi.org/10.15407/mag17.04.441

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