This paper focuses on the equational class S„ of Brouwerian algebras and the equational class L„ of Heyting algebras generated by an »-element chain. Firstly, duality theories are developed for these classes. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in Sn and L„ are characterized. Finally, free prod...

There are two standard model-theoretic methods for proving the finite model property for modal and superintuionistic logics, the standard filtration and the selective filtration. While the corresponding algebraic descriptions are better understood in modal logic, it is our aim to give similar algebraic descriptions of filtrations for superintuitionistic logics via locally finite reducts of Heyt...

In this paper we introduce the notion of generalized implication for lattices, as a binary function⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized im...

Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a proof-theoretic point of view, deduction modulo defines a generic notion of cut that applies to any first-order theory presented as a rewrite system. In such a setting, one can prove cut-elimination theorems that apply to many theo...

A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are de...

Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new s...

We show that a large number of equations are preserved by DedekindMacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that a large class of varieties of Heyting algebras and FL-algebras admits completions.

Based on the paper [4] we show that true second-order arithmetic is interpretable over real-algebraic structure of models intuitionistic analysis built upon a certain class complete Heyting algebras.