‎fuzzy subgroups and digraphs induced by fuzzy subgroups

Given a fuzzy subgroup µ of a group g, x ▷u y if and only if µ(xy) < µ(yx) defines a directed relation with an associated digraph (g, ▷u). we consider (µ, ν)-homomorphisms φ : (g, µ) → (h, ν) where µ and ν are fuzzy subgroups of g and h respectively and the preservation of properties of the digraphs (g, ▷u) several of which are also noted here, e.g., (g, ▷u) is an anti-chain if and only if µ is a fuzzy normal subgroup of the group g.