Homomorphism Preserving Algebraic Specifications Require Hidden Sorts

Abstract

Although every computable data type has an initial algebra specification with hidden functions, it may happen that some of the homomorphic images of the data type are not models of the specification. The latter are reducts of algebras that would be models of the specification if all its functions were visible, whereas the homomorphic images of the data type are independent of the specification and need not be compatible with the hidden functions used in it. A hidden function specification that does not exclude any of the homomorphic images of its initial model from its model class will be called homomorphism preserving . It turns out that, unlike unrestricted initial homomorphism preserving. It turns out that, unlike unrestricted initial algebra specification, homomorphism preserving initial algebra specification of computable data types requires both hidden sorts and hidden functions.