Profinite bi-Heyting algebras

Abstract A poset $${\mathbb {X}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> 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 $$x\in X.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> 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.