SN Combinators and Partial Combinatory Algebras

We introduce an intersection typing system for combinatory logic, such that a term of combinatory logic is typeable ii it is sn. We then prove the soundness and completeness for the class of partial com-binatory algebras. Let F be the class of non-empty lters which consist of types. Then F is an extensional non-total partial combinatory algebra. Furthermore, it validates the strongest consistent equality of the set of sn terms of combinatory logic. By F, we can solve Bethke-Klop's question; \\nd a suitable representation of the nally collapsed partial combinatory algebra of P". Here, P is a partial combinatory algebra, and is the set of closed sn terms of combinatory logic modulo the inherent equality. Our solution is the following: the nally collapsed partial combinatory algebra of P is representable in F. To be more precise, it is isomorphically embeddable into F.