Free algebras and coproducts in varieties of G\"odel algebras

TL;DR

This paper provides a dual space description of free G"odel algebras and coproducts using Priestley and Esakia dualities, extending known results to arbitrary generators and varieties.

Contribution

It generalizes dual descriptions of free G"odel algebras and coproducts to infinite generators and all varieties, with explicit formulas and properties.

Findings

Dual description of free G"odel algebras via Esakia spaces.

Characterization of coproducts of G"odel algebras using dual spaces.

Free G"odel algebras are bi-Heyting algebras.

Abstract

G\"odel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free G\"odel algebras and coproducts of G\"odel algebras. In particular, we realize the Esakia space dual to a G\"odel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free G\"odel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of G\"odel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of G\"odel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of G\"odel algebras and show that all free G\"odel algebras are bi-Heyting algebras.