On the existence of extensional partial combinatory algebras

Journal of Symbolic Logic 52 (3):819-833 (1987)
The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in $\S2$ . In $\S0$ 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 $\S1$ we discuss some properties of nontotal extensional combinatory algebras in general. $\S2$ describes a "partial" variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, $\S3$ deals with properties of the models constructed in $\S2$ , such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms
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