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We conceive a parametrized data type as a partial functor @: ALG (Z) ~ ALG (A), where A is a signature extending Z and ALG (Z) is the class of minimal S-algebras which serve as parameters. We focus attention on one particular method of algebraically specifying parametrized data types: finite specifications with conditional equations using auxiliary sorts and functions provided with initial algebra semantics. We introduce the concept of an effective parametrized data type. A satisfactory adequacy result is then obtained: each effective parametrized data type possesses a finite algebraic specification under initial semantics.

Algebraic specifications for parametrized data types with minimal parameter and target algebras