  The present paper extends generalized morphisms of relations into the realm
of Monoidal Fuzzy Logics by first proving and then using relational
inequalities over pseudo-associative BK-products (compositions) of relations in
these logics.
  In 1977 Bandler and Kohout introduced generalized homomorphism,
proteromorphism, amphimorphism, forward and backward compatibility of
relations, and non-associative and pseudo-associative products (compositions)
of relations into crisp (non-fuzzy Boolean) theory of relations. This was
generalized later by Kohout to relations based on fuzzy Basic Logic systems
(BL) of H\'ajek and also for relational systems based on left-continuous
t-norms.
  The present paper is based on monoidal logics, hence it subsumes as special
cases the theories of generalized morphisms (etc.) based on the following
systems of logics: BL systems (which include the well known Goedel, product
logic systems; Lukasiewicz logic and its extension to MV-algebras related to
quantum logics), intuitionistic logics and linear logics.
