Profinite bi-Heyting algebras

Abstract

A poset \({\mathbb {X}}\) is said to be zigzag image-finite, if the least updownset (i.e., both an upset and a downset) containing x is finite, for all \(x\in X.\) We show that a bi-Heyting algebra is profinite if and only if it is isomorphic to the lattice of upsets of a zigzag image-finite poset. Zigzag image-finite posets have the property of being disjoint unions of finite connected posets. Because of this, we equivalently show that a bi-Heyting algebra is profinite if and only if it is isomorphic to a direct product of simple finite bi-Heyting algebras.