Profinite Heyting algebras (Q953270)

A Heyting algebra \(A\) is called completely join-prime generated if every element of \(A\) is a join of completely join-prime elements of \(A\). Theorem 2.12 of this paper characterizes complete and completely join-prime generated Heyting algebras in two different ways. An algebra \(A\) is called profinite if it is isomorphic to the inverse limit of an inverse system of finite algebras. Theorem 3.6 provides five conditions, each of them characterizing profinite Heyting algebras. Several consequences of this theorem and related results are also obtained. These include Theorem 4.2, which states that the linear Heyting algebras \(L_{\infty}\) and \(L_n\) are the only profinite linear Heyting algebras, and Theorem 4.4, which characterizes profinite bounded distributive lattices.