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An algebraic view of recursive definitions is presented, extending an already familiar analogy with homomorphisms. A notion of simulation of one recursive definition by another is then defined. This leads to a particular approach to verification and transformation, which places emphasis on the arrows between programs, rather than the programs themselves. These arrows are the program morphisms of the title. Examples are given, together with certain extensions of the idea. Also indicated is a methodology which can lead to the discovery of program morphisms and new equivalent versions of a given program. [ABSTRACT FROM AUTHOR]