On frontal operators in Hilbert algebras

Abstract

In this article, we introduce and study a family of compatible functions in Hilbert algebras which in the case of Heyting algebras agree with the frontal operators given by Esakia (2006, J. Appl. 16,[349][350][351][352][353][354][355][356][357][358][359][360][361][362][363][364][365][366]. Moreover, we give a representation theory, based on previous works by Cabrer, Celani and Montangie, for Hilbert algebras with a frontal operator and for Hilbert algebras with some particular frontal operators.

On the free frontal implicative semilattice extension of a frontal Hilbert algebra

“…The simplest examples of compatible functions in an algebra A are the polynomial functions. 2 Frontal operators on Heyting algebras are necessarily compatible functions as well as the frontal operators on Hilbert algebras are, as it was proved in [7]. Definition 7.…”

“…The following definition was introduced in [7] and generalizes the definition of frontal operator given for Heyting algebras to the Hilbert algebras setting. Definition 6.…”

“…In [7] the notion of γ-operator was generalized to the framework of bounded Hilbert algebras. As in Heyting algebras, if H ∈ Hil 0 and a ∈ H we define ¬a := a → 0.…”

“…But γoperators may not exist. In [7,Section 3,Example 14], in contrast with the case of Heyting algebras, we find examples of finite bounded Hilbert algebras that lack the γ-operator. When a γ-operator exists, it will be denoted, as we did for Heyting algebras, by γ.…”

“…In [7] the notion of frontal operator is generalized to Hilbert algebras, and hence to implicative semilattices too, as well as the frontal operators γ, S and G considered by Caicedo and Cignoli in [4] (examples 3.1, 5.2 and 5.3 respectively) in the framework of Heyting algebras as particular examples of implicitly definable compatible functions.…”