Combinatory algebras (defined below) provide a useful framework for studying functions and functional application sinct algebraically definable functions over the combinatory algebra are representable (combinatory comp!eteness). Unfortunately, these structures are not in general rich enough to provide a sound interpretation for the untyped A-calculus. In particular, rule (5) : u = u--hx.u = Ax. 2;. is not valid in all combinatoly algebras. Thus strdnger structures, called /i-models, are needed to intcpret the A-calculus. Since the original inverse limit construction by Scott. several A-models have been defined. In particular, Scott [ 131 has also defined another model, (Pm, l , !P), over the set of all subsets of the natural numbers. Scott’!: ide.!s have been developed in several directions (see [8,1,7,4, 11,2,3] for applications to constructive set theory, recursion theory in higher types, and for model-tht,oretic results). In Stoy [14] there is a rather detailed presentation of the main area of application of the semantics of A-calculus to theoretical computer science: the den,ltational semantics of programming languages. While not all combinatory algebras can be made into A-models, many can. in this paper we examine such expansions of combinatory algebras. Most of the focus here is on Scott’s PO model, where we show that there is a unique expansion of the applicative structure (Pm, .) to a A-model. We also show the uniqueness of k’