A Sufficient Condition for Completability of Partial Combinatory Algebras

Abstract

A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 11, posed by D. Scott, H. Barendregt, and G. Mitschke) if every PCA can be completed. A negative answer to this question was given by Klop in [12, 11]; moreover he provided a sufficient condition for completability of a PCA $(M, \cdot, K, S)$ in the form of ten axioms (inequalities) on terms of $M$. We prove that just one of these axiom (the so called Barendregt's axiom) is sufficient to guarantee (a slightly weaker notion of) completability.