The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do (...) not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (shrink)

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

We study the computational complexity of the universal and quasi-equational theories of classes of bounded distributive lattices with a negation operation, i.e., a unary operation satisfying a subset of the properties of the Boolean negation. The upper bounds are obtained through the use of partial algebras. The lower bounds are either inherited from the equational theory of bounded distributive lattices or obtained through a reduction of a global satisfiability problem for a suitable system of propositional modal logic.

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables, upon the basis of the conception of model introduced in :27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models that a rule is derivable in (...) the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics. (shrink)

In this paper we introduce the class of weak Heyting–Brouwer algebras (WHB-algebras, for short). We extend the well known duality between distributive lattices and Priestley spaces, in order to exhibit a relational Priestley-like duality for WHB-algebras. Finally, as an application of the duality, we build the tense extension of a WHB-algebra and we employ it as a tool for proving structural properties of the variety such as the finite model property, the amalgamation property, the congruence extension property and the (...) Maehara interpolation property. (shrink)

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.

In this paper we study the variety $\textsf{WL}$ of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the $\{\vee,\wedge,\Rightarrow,\bot,\top \}$ -fragment of the arithmetical base preservativity logic $\mathsf {iP^{-}}$. The variety $\textsf{WL}$ properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means (...) of WL-frames. We extended this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic $\textsf{iP}^{-}$. (shrink)

Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters and tense centered deductive (...) systems, respectively. (shrink)

It is well known that for all recursively enumerable sets X 1 , X 2 there are disjoint recursively enumerable sets Y 1 , Y 2 such that $Y_1 \subseteq X_1, Y_2 \subseteq X_2$ and Y 1 ∪ Y 2 = X 1 ∪ X 2 . Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated (...) distributive lattices is undecidable. (shrink)

The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated (...) modal logic. This suggests that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

Using known facts we give a simple characterization of the distributivity of lattices of finite length.

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely (...) axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics (...) that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. (shrink)

In this paper, we prove that the lattice of all closure operators of a complete Almost Distributive Lattice L with fixed maximal element m is dual atomistic. We define the concept of a completely meet-irreducible element in a complete ADL and derive a necessary and sufficient condition for a dual atom of Φ (L) to be complemented.

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety (...) of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models. (shrink)

The category $\mathbb {DRDL}{'}$, whose objects are c-differential residuated distributive lattices satisfying the condition $\textbf{CK}$, is the image of the category $\mathbb {RDL}$, whose objects are residuated distributive lattices, under the categorical equivalence $\textbf{K}$ that is constructed in Castiglioni et al. (Stud Log 90:93–124, 2008). In this paper, we introduce weak monadic residuated lattices and study some of their subvarieties. In particular, we use the functor $\textbf{K}$ to relate the category $\mathbb {WMRDL}$, whose objects are weak monadic residuated (...) distributive lattices, and the category $\mathbb {WMDRDL}{'}$, whose objects are pairs formed by an object of $\mathbb {DRDL}{'}$ and a center weak universal quantifier. (shrink)

We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice.

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

The present note is an Erratum for the two theorems of the paper "Congruences and ideals in a distributive lattice with respect to a derivation" by M. Sambasiva Rao.

Gödel algebras form the locally finite variety of Heyting algebras satisfying the prelinearity axiom =. In 1969, Horn proved that a Heyting algebra is a Gödel algebra if and only if its set of prime filters partially ordered by reverse inclusion–i.e. its prime spectrum–is a forest. Our main result characterizes Gödel algebras that are free over some finite distributive lattice by an intrisic property of their spectral forest.

Peirce introduced Existential Graphs in late 1896, and they were systematically investigated in his 1903 Lowell Lectures. Alpha graphs for classical propositional logic constitute the first part of EGs. The second and the third parts are the beta graphs for first-order logic and the gamma graphs for modal and higher-order logics, among others. As a logical syntax, EGs are two-dimensional graphs, or diagrams, in contrast to the linear algebraic notations. Peirce's theory of EGs is not only a theory (...) of logical syntax but also a proof theory for many logics. A set of transformation rules of graphs defines a proof system for a graphical logic. In this... (shrink)

The notion of a conditionally distributive lattice was introduced by B. Wolniewicz while formally investigating the ontology of situations . In several of this lectures he has appealed for a study of that class of lattices. The present abstract is a response to that request.

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. There are some problems related to the motivation of the operation of Boolean complementation. Because of this operation is dropped and the language of distributive lattices is extended by considering as non-definable primitives the relations of contact, nontangential inclusion and dual contact. It is obtained an axiomatization of the (...) theory consisting of the universal formulas in the language true in all contact algebras. The structures in, satisfying the axioms in question, are called extended distributive contact lattices. In this paper we consider several logics, corresponding to EDCL. We give completeness theorems with respect to both algebraic and topological semantics for these logics. It turns out that they are decidable. (shrink)

In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA (...) and of all distributive closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimoto's characterization of distributive lattices, and Priestley's topological representation of the congruence lattice of a bounded distributive lattice. (shrink)

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice (...) logic with n -ary additive or multiplicative operators, with negation operators modally interpreted as impossibility and unnecessity, as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard interpretation of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. (shrink)

Arising in the study of Quantum Logics, PBZ $^{*}$ -_lattices_ are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition with respect to the Brouwer complement. They form a variety $\mathbb {PBZL}^{*}$ which includes that of orthomodular lattices considered with an extended signature (by endowing them with a Brouwer complement coinciding with their Kleene complement), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have (...) directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements, since the elements with bounded lattice complements coincide to the sharp elements in distributive PBZ $^{*}$ -lattices. The variety $\mathbb {DIST}$ of distributive PBZ $^{*}$ -lattices has an infinite ascending chain of subvarieties, and the variety generated by orthomodular lattices and antiortholattices has an infinity of pairwise disjoint infinite ascending chains of subvarieties. (shrink)

Arising in the study of Quantum Logics, PBZ $$^{*}$$ -lattices are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition with respect to the Brouwer complement. They form a variety $$\mathbb {PBZL}^{*}$$ which includes that of orthomodular lattices considered with an extended signature (by endowing them with a Brouwer complement coinciding with their Kleene complement), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have (...) directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements, since the elements with bounded lattice complements coincide to the sharp elements in distributive PBZ $$^{*}$$ -lattices. The variety $$\mathbb {DIST}$$ of distributive PBZ $$^{*}$$ -lattices has an infinite ascending chain of subvarieties, and the variety generated by orthomodular lattices and antiortholattices has an infinity of pairwise disjoint infinite ascending chains of subvarieties. (shrink)