single-combinator basis with direct algebraic characterisation?

From: r.e.s. (r.s_at_ZZmindspring.com)
Date: 11/01/04 

Date: Mon, 01 Nov 2004 21:28:37 GMT

Combinators K,S can be characterised as satisfying
   K a b = a
   S a b c = a c (b c)
for arbitrary terms a,b,c.

Does there exist a single-combinator basis, say X,
which can be similarly characterised as satisfying
   X a b c ... = ...
in which only the arbitrary terms a,b,c,... appear
on the RHS? (X := \f.fKSK fails, evidently.)

--r.e.s.

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