A sufficient condition for completability of partial combinatory algebras

A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 1 1, 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, ., 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. ?