Profinite Heyting algebras and the representation problem for Esakia spaces

Theory and Logic Group hosting a talk by Tommaso Moraschini

ABSTRACT

A poset is said to be "representable" if it can be endowed with an Esakia topology. Gratzer's classical representation problem asks for a description of representable posets. A solution to this problem is however not expected to take a simple form, as representable posets do not form an elementary class. Since at the moment a solution to the representation problem seems out of reach, we will address a simpler version of the problem which, roughly speaking, asks to determine the posets that may occur as top parts of Esakia spaces. Crossing the bridge between algebra and topology, this task amounts to characterizing the profinite Heyting algebras that are also profinite completions of some Heyting algebras.

Accordingly, in this talk we shall present an exhaustive characterization of varieties of Heyting algebras whose profinite members are profinite completions. The structure theory of these varieties is indeed very rich. Not only all of them are locally finite and finitely axiomatizable, but also they are hereditarily structurally complete and have the Beth definability property. Connection with the representation problem and decidability will be also discussed at the end of the talk.

This talk is based on joint work with G. Bezhanishvili, N. Bezhanishvili, and M. Stronkowski.

Tommaso Moraschini, University of Barcelona