Weak Implication on Monadic Heyting Algebras

Abstract

In this paper, we introduce an implication operation, called weak implication, which will be quite useful in order to characterize subdirectly irreducible monadic Heyting algebras. Furthermore, it is shown that deductively semisimple algebras are the non trivial ones such that the subalgebra of constants is a Tarski algebra with rst element, i.e. a Boolean algebra, as it is mentioned by A. Monteiro and O. Varsavsky in 1957 (Algebras de Heyting monádicas, Actas de las X Jornadas de la Unión Matemática Argentina, Bahía Blanca, (1957), (52-62). Finally, it is stated that some of the results established for monadic Heyting algebras are also valid for monadic generalized Heyting algebras.