Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 4, pp. 441-462   https://doi.org/10.15407/mag17.04.441     ( to contents , go back )

### Membership Deformation of Commutativity and Obscure n-ary Algebras

Steven Duplij

Center for Information Technology (WWU IT), Universität Münster, D-48149 Münster, Deutschland
E-mail: douplii@uni-muenster.de, sduplij@gmail.com

Received September 17, 2020, revised April 15, 2021.

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.

Mathematics Subject Classification 2010: 16U80, 20C35, 20N15, 20N25
Key words: almost commutative algebra, obscure algebra, membership deformation, fuzzy set, membership function, n-ary algebra, Lie algebra, projective representation