We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar $\mathbf{BI}(\_)^\bullet$-algebras as well as the braided $\mathbf{BC^\pm I}$-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.

翻译：本文论证了操作域为处理组合代数中的多项式和组合完备性提供了一个通用框架，涵盖经典的$\mathbf{SK}$-代数、线性$\mathbf{BCI}$-代数、平面$\mathbf{BI}(\_)^\bullet$-代数以及辫状$\mathbf{BC^\pm I}$-代数。我们证明每个外延组合代数都能导出一个典范闭操作域，称之为该组合代数的内部操作域。内部操作域构造给出了从闭操作域到外延组合代数的遗忘函子的左伴随。作为副产品，我们推导了上述几类组合代数的外延公理。