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：我们认为操作代数提供了一个通用框架，可用于处理组合代数的多项式及组合完备性问题，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生成一个典范的闭操作代数，并将其称为该组合代数的内部操作代数。内部操作代数构造给出了从闭操作代数到外延组合代数的遗忘函子的左伴随。作为副产品，我们推导出了上述几类组合代数的外延性公理。