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

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 $\epsilon$-commutativity are introduced in such a way that equal degrees of truth result in the “nondeformed” case. We also sketch how to “deform” $\epsilon$-Lie algebras and Weyl algebras. Further, the above constructions are extended to $n$-ary algebras for which the projective representations and $\epsilon$-commutativity are studied.