  We introduce an equationally definable counterpart of the notion of
combinatory model. The new notion, called an algebraic combinatory model, is
weaker than that of a lambda algebra but is strong enough to interpret lambda
calculus. The class of algebraic combinatory models admits finite
axiomatisation with seven closed equations, and it can be characterised as the
retracts of combinatory models. Lambda algebras are then characterised as
algebraic combinatory models which are stable, and there is a canonical
construction of a lambda algebra from an algebraic combinatory model. This
passage to a lambda algebra also manifests itself in our construction of a
cartesian closed category with a reflexive object from an algebraic combinatory
model. The resulting axiomatisation of lambda algebras with the seven equations
and stability corresponds to that of Selinger [J. Funct. Programming, 12(6),
549-566, 2002], which would clarify the origin and the role of each axiom in
his axiomatisation.
