Variations of the free implicative semilattice extension of a Hilbert algebra

TL;DR

This paper offers an alternative construction for the free implicative semilattice extension of Hilbert algebras, establishes an adjunction between related algebraic categories, and introduces a potentially useful functor connecting these structures.

Contribution

It provides a new method for constructing free extensions, demonstrates an adjunction between Hilbert algebras with supremum and generalized Heyting algebras, and defines a novel functor between these categories.

Findings

Alternative construction for free implicative semilattice extension.

Establishment of an adjunction between algebraic categories.

Introduction of a functor from Hilbert algebras to generalized Heyting algebras.

Abstract

In [{\it On the free implicative semilattice extension of a Hilbert algebra}. Mathematical Logic Quarterly 58, 3 (2012), 188--207], Celani and Jansana give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in last section we describe a functor from the algebraic category of Hilbert algebras to that of generalized Heyting algebras, of possible independent interest.