In this note, we prove that having unique head-normal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CL-terms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras.

Mathematical Logic (acm F.4.1)
Partial algebras (msc 08A55), Combinatory logic and lambda-calculus (msc 03B40), Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) (msc 16S15)
CWI
Department of Computer Science [CS]
Specification and Analysis of Embedded Systems

Bethke, I, Klop, J.W, & de Vrijer, R. (1995). Completing partial combinatory algebras with unique head-normal forms. Department of Computer Science [CS]. CWI.