Sesión Lógica y ComputabilidadA Modal Expansion of Kleene Algebras via Twist Structures
Valeria Miguelez
Universidad Nacional del Centro de la Provincia de Buenos Aires (UNICEN), Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
Jalali’s representation of Kleene algebras by means of triples \(\langle D, R, F \rangle\), where \(D\) is a bounded distributive lattice, \(R\) is an independence relation dual to contact, and \(F\) is a Boolean filter [1], yields a categorical equivalence between Kleene algebras and a category of triples. In this talk we study how this representation can be enriched by incorporating modal operators on the distributive lattice component. This leads to the notion of modal triples, extending the framework of Kleene triples in a way analogous to the modal representation recently introduced for Nelson algebras [2]. We show that modal triples yield a class of modal Kleene algebras and establish a categorical equivalence between these two structures.
The modal triple framework also provides a natural setting for studying implication. Motivated by the fact that, in Heyting algebras, every implication determines a pair of modal operators, we show that an implication on the lattice component gives rise to a parametric family of modal operators. As a consequence, the modal triple framework provides a representation of Kleene algebras with implication, yielding a categorical equivalence between lattices with implication and Kleene algebras equipped with implication.
Trabajo en conjunto con: Paula Menchón (UNICEN) y Sergio Celani (UNICEN).
Referencias
[1] Jalali, A. A., A representation of Kleene algebras and the characterisation of projective Kleene and De Morgan algebras, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, No. 29 (1992), 435–474.
[2] Menchón, P. and Rodriguez, R. O., Twist-structures isomorphic to modal Nelson lattices, arXiv:2504.08109, 2025.