Summary: We define an \(\epsilon\)-fuzzy congruence, which is a weakened fuzzy congruence, find the \(\epsilon\)-fuzzy congruence generated by the union of two \(\epsilon\)-fuzzy congruences on a semigroup, and characterize the \(\epsilon\)-fuzzy congruences generated by fuzzy relations on semigroups. We also show that the collection of all \(\epsilon\)-fuzzy congruences on a semigroup is a complete lattice and that the collection of \(\epsilon\)-fuzzy congruences under some conditions is a modular lattice.