The Internal Operads of Combinatory Algebras

11/20/2022

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by   Masahito Hasegawa, et al.

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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.