Masahito Hasegawa - The Internal Operads of Combinatory Algebras

entics:10338 - Electronic Notes in Theoretical Informatics and Computer Science, February 22, 2023, Volume 1 - Proceedings of MFPS XXXVIII - https://doi.org/10.46298/entics.10338
The Internal Operads of Combinatory AlgebrasArticle

Authors: Masahito Hasegawa

    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.


    Volume: Volume 1 - Proceedings of MFPS XXXVIII
    Published on: February 22, 2023
    Accepted on: November 22, 2022
    Submitted on: November 22, 2022
    Keywords: Computer Science - Logic in Computer Science,03B40 (Primary) 68Q55, 03B47, 18C50, 18M65 (Secondary),F.3.2

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