Since Milner and Park’s work on equivalence of CCS agents, co-inductively defined notions of program equivalence have become as indispensable tool in the study of concurrent processes. Abramsky and Ong’s work on the lazy lambda calculus, work of the Cornell school (Allen, Howe, Smith) on the semantics of NuPrl, and Andrew Gordons Ph.D. thesis show that such co-inductive notions of program ... [Show full abstract] equivalence and program ordering can be developed for functional programming languages and are useful. In particular, as a consequence of Howe’s general congruence result such notions of ‘applicative (bi)similarity’ are sound for establishing Morris-style contextual equivalence of programs. For deterministic, pure functional languages the two notions of equivalence may coincide, although this depends rather delicately upon the notion of observation and the expressiveness of the language constructs. In this talk I discuss a property of applicative similarity that expresses every program as the least upper bound of a chain of syntactically defined projections, and which is the analogue of the ‘minimal invariant’ property of recursively defined domains. Although quite technical in nature, the property is a key one for developing many domain-theoretic properties of an operational semantics. This syntactic version of minimal invariance has been studied in detail by Scott Smith. Here I sketch a simplified proof and give one simple application of the result – proving ‘rational completeness’ for Church’s fix-point combinator up to applicative bisimilarity.