Functional monadic Heyting algebras.
A Heyting algebra is a bounded distributive lattice with the operation of relative pseudocomplementation. A monadic Heyting algebra is a Heyting algebra equipped with two unary operations \(\Delta \) and \(\nabla \) satisfying the conditions (i) \(\Delta \nabla a = \nabla a\) (ii) \(\nabla \Delta a = \Delta a\) (iii) \(\nabla (\nabla a\wedge b) = \nabla a \wedge \nabla b\). It forms an algebraic counterpart of intuitionistic predicate calculus where \(\Delta \) or \(\nabla \) are required as the intuitionistic quantifiers \(\forall \) or \(\exists \), respectively. For a complete Heyting algebra \(H\) and a set \(X\), define operations \(\Delta , \nabla \) on the Heyting algebra \(H^X\) of all functions \(f: X\to H\) by setting \[ (\Delta f)(x) = \bigwedge \{ f(y);\;y\in X \} \quad \text{and} \quad (\nabla f)(x) = \bigvee \{ f(y);\;y\in X \}. \] The resulting algebra is called a functional Heyting algebra. It was a problem of A. Monteiro and O. Varsavsky whether every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. The authors present a new method for solving this problem in the positive.
scientific article; zbMATH DE number 2158793