This paper presents a semantics for a simple language that is a blend of algebraic models and traditional denotational semantics. In this semantics, implementations of user-defined abstract data types are "compiled" into an algebraic structure, which is used by the denotational part of the semantics whenever an operation of the data type is invoked. To show the utility of such a semantics, an algebraic characterization of simulation between states over such algebras is given, and it is shown that simulation is preserved by expressions and commands in the language. (Note: versions TR93-21 and TR93-21a were titled: "A Model Theory for Abstract Data Types with Mutable Objects (extended abstract)".)