The Internal Operads of Combinatory Algebras

We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical SK-algebras, linear BCI-algebras, planar BI(₎^•-algebras as well as the braided BC^± I-algebras. We show that every extensional combinatory algebra gives rise to a canonical closed operad, which we shall call the internal operad of the combinatory algebra. The internal operad construction gives a left adjoint to the forgetful functor from closed operads to extensional combinatory algebras. As a by-product, we derive extensionality axioms for the classes of combinatory algebras mentioned above.