Variations of the free implicative semilattice extension of a Hilbert algebra

Abstract

In [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.

Prelinear Hilbert algebras

“…We use it in order to give an explicit description of the left adjoint of the forgetful functor from the algebraic category of prelinear Heyting algebras to the algebraic category of bounded prelinear Hilbert algebras. The ideas used in this section are similar to that developed in [7]. In Section 4 we apply results of the previous section in order to study some descriptions of the coproduct of two finite algebras in the algebraic category of bounded prelinear Hilbert algebras.…”

“…is an implicative semilattice, which is called the free implicative semilattice extension of the Hilbert algebra H (see also [7]). Proof.…”

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

“…There are several ways to obtain such a left adjoint. Explicit descriptions of such an adjoint functor and of the free extension of a Hilbert algebra to an implicative semilattice are obtained in [10] and [8].…”

“…For completeness of the exposition, we provide now a description of an adjoint functor to the forgetful functor from IS to Hil, which will be used later to obtain our results. We give complete proofs using only the minimum tools needed to obtain the results, thus avoiding more general approaches such that those in [10,8].…”

“…In [10] an explicit definition of a left adjoint of the forgetful functor from the category of implicative semilattices to the category of Hilbert algebras is given. In [8] another explicit description of such an adjoint functor was presented following an alternative path.…”