  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.
