Profiniteness and representability of spectra of Heyting algebras

TL;DR

This paper investigates the conditions under which profinite Heyting algebras are isomorphic to profinite completions, resolving an open problem and characterizing representable varieties with implications for the representation of posets.

Contribution

It proves the existence of profinite Heyting algebras not isomorphic to any profinite completion and characterizes varieties where such isomorphisms occur, including a largest such variety.

Findings

Existence of non-isomorphic profinite Heyting algebras to any profinite completion.

Characterization of varieties with isomorphic profinite algebras and completions.

Decidability of whether all profinite members are completions in finitely axiomatizable varieties.

Abstract

We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are profinite completions. In addition, we introduce and characterize representable varieties of Heyting algebras, thus drawing connection to the classical problem of representing posets as prime spectra.