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Realizability in ordered combinatory algebras with adjunction

Published online by Cambridge University Press:  26 April 2018

WALTER FERRER SANTOS ,
MAURICIO GUILLERMO  and
OCTAVIO MALHERBE
WALTER FERRER SANTOS
Affiliation:
Departamento de Matemática y Aplicaciones, Centro Universitario Regional del Este, Universidad de la República, Tacuarembó entre Avenida Artigas y Aparicio Saravia, Maldonado, Uruguay Email: wrferrer@cmat.edu.uy Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225 11400, Montevideo, Uruguay Email: mguille@fing.edu.uy
MAURICIO GUILLERMO
Affiliation:
Instituto de Matemática y Estadísitica Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565 11300, Montevideo, Uruguay Email: malherbe@fing.edu.uy
OCTAVIO MALHERBE
Affiliation:
Departamento de Matemática y Aplicaciones, Centro Universitario Regional del Este, Universidad de la República, Tacuarembó entre Avenida Artigas y Aparicio Saravia, Maldonado, Uruguay Email: wrferrer@cmat.edu.uy Instituto de Matemática y Estadísitica Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565 11300, Montevideo, Uruguay Email: malherbe@fing.edu.uy
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Abstract

In this work, we continue our consideration of the constructions presented in the paper Krivine's Classical Realizability from a Categorical Perspective by Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented, mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 3 , March 2019 , pp. 430 - 464
Copyright
Copyright © Cambridge University Press 2018 

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