Combinatory foundation of functional programming

A proposal is described for embedding FP and a part of FFP into a system <italic>C</italic>(<italic>IN</italic>)<italic>V</italic> of Combinatory Logic generated by the 6-tuple of combinators (A,B,C<subscrpt>*</subscrpt>,O,L,D) under the operation of application. At the same time <italic>C</italic>(<italic>IN</italic>)<italic>V</italic> is viewed as an algebraic extension of elementary arithmetic, including addition multiplication and exponentiation, leading to a non commutative semi-ring with an infinity of zero (infinite)-like elements. Two interesting submonoids have been selected: <italic>L</italic><subscrpt>º</subscrpt> able to represent the set of FP-Sequences and <italic>L</italic><subscrpt>+</subscrpt> able to represent the set of FP-Constructions. All basic objects forming operators like Composition, Apply To All Condition and Insert Right are then efficiently expressed inside <italic>C</italic>(<italic>IN</italic>)<italic>V</italic>. The same is done for some operators belonging to FFP as Lifting and the APPLY of LISP establishing a basis for a future reduction (operational) semantics of FP.