Vague Groups

Abstract

In the crisp case, if (G, ∘ ) is a set with an operation ∘ : G ×G →G and ~ is an equivalence relation on G, then ∘ is compatible with ~ if and only if

$$ a \sim a' \ {\rm and} \ b \sim b' \ {\rm implies} \ a \circ b \sim a' \circ b'. $$

In this case, an operation $\tilde{\circ}$ can be defined on $\overline{G}=G/\sim$ by

$$ \overline{a} \tilde{\circ} \overline{b}=\overline{a \circ b} $$

where $\overline{a}$ and $\overline{b}$ are the equivalence classes of a and b with respect to ~.

Demirci generalized this idea to the fuzzy framework by introducing the concept of vague algebra, which basically consists of fuzzy operations compatible with given indistinguishability operators [37].