EQ-algebras from the point of view of generalized algebras with fuzzy equalities

EQ-algebras introduced by Novák are algebras of truth values for a higher-order fuzzy logic (fuzzy type theory). In this paper, the compatibility of multiplication w.r.t. the fuzzy equality in an arbitrary EQ-algebra is examined. Particularly, an example indicates that the compatibility axiom does not always hold, and then a class of EQ-algebras satisfying the compatibility axiom is characterized by introducing a residuated integral-meet-semilattice-ordered-monoid-valued fuzzy algebra called a generalized algebra with a fuzzy equality.