On the existence of extensional partial combinatory algebras

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Abstract
The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in §2. In §0 we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In §1 we discuss some properties of nontotal extensional combinatory algebras in general. §2 describes a “partial” variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, §3 deals with properties of the models constructed in §2, such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms.

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