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:我们争论说,歌剧为处理复合代数的多元性和组合性完整性提供了一个总体框架,包括古典 $mathbf{SK}$-algebras、线性 $mathbf{BCI}$BCI}-algebras、平板 $mathbf{BI}( ⁇ ) ⁇ bullet$-algebras以及编织 $mathbf{BC ⁇ pm I}-algebras。我们表明,每个扩展的复燃代数都会产生一个罐形闭合的手术,我们称之为复燃代数的内演。内部剧组构造让从封闭的歌剧到扩展的复燃性代数的遗忘的喜好者左接。作为副产品,我们为上文提及的复燃性代数的类别产生扩展性轴轴。