An infinitary extension of Jankov's theorem (Q2454639)

\textit{V. A. Jankov}'s [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 18--38 (1969; Zbl 0181.00404)] celebrated result states that for every finite subdirectly irreducible Heyting algebra \(A\), there exists a formula \(\chi_A\), called the Jankov formula of \(A\), such that a Heyting algebra \(B\) refutes \(\chi_A\) iff \(A\) is embeddable into a quotient of \(B\). In this paper, the author presents an infinitary version of Jankov's theorem. A Heyting algebra homomorphism between two complete Heyting algebras is called continuous if it preserves all joins and meets. The author introduces a notion of \(T\)-regular complete Heyting algebras and pseudo-continuous homomorphisms between complete Heyting algebras and proves that for every subdirectly irreducible Heyting algebra \(A\), there exists a formula \(\chi_A\) in the infinitary language such that a complete \(T\)-regular Heyting algebra \(B\) refutes \(\chi_A\) iff there exists a complete Heyting algebra \(D\), a continuous embedding of \(A\) into \(D\) and a pseudo-continuous homomorphism from \(B\) onto \(D\).