Sergio Celani

The variety of weak Heyting algebras was introduced in 2005 by Celani and Jansana. This corresponds to the strict implication fragment of the normal modal logic K$K$ which is also known as the subintuitionistic local consequence of the class of all Kripke models. Subresiduated lattices are a generalization of Heyting algebras and particular cases of weak Heyting algebras. They were introduced during the 1970s by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. In this paper we study the class of implicative‐infimum subreducts of weak Heyting algebras. In particular, we prove that this class is a variety by giving an equational base for it. We also present a topological duality for the algebraic category whose objects are the implicative‐infimum subreducts of subresiduated lattices.

In this paper, we introduce and study the variety of algebras (A,∧,∨,→,◻,◊,0,1) of type (2,2,2,1,1,0,0) whose {∧,∨,→,0,1}-reduct is a weak Heyting algebra and the following identities are satisfied: (a) ◻1=1, (b) ◻(a∧b)=◻a∧◻b, (c) ◊0=0 and (d) ◊(a∨b)=◊a∨◊b. This variety, which is denoted by MWH, contains several varieties of Heyting algebras with modal operators, which are the algebraic semantics of well-known modal intuitionistic logics. The main goal of this paper is to study certain subvarieties of MWH, which are the algebraic semantics of particular subintuitionistic modal logics. We give representation theorems for these subvarieties, we prove that they are canonical and we also show that their associated deductive systems are frame complete with respect to certain class of relational frames.

Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \(\left(A,f,g\right)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)\leq b\) iff \(a\leq g(b)\), and \(g(a\rightarrow b)\leq g(a)\rightarrow g(b)\) forall \(a,b\in A\). In this paper, we are going to prove that some varieties of HilGC-algebras are characterized by first-order conditions defined in the dual space and that these varieties are canonical. Additionally, we will also study and characterize the congruences of an HilGC-algebra through specific closed subsets of the dual space. This characterization will be applied to determine the simple algebras and subdirectly irreducible HilGC-algebras.

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.

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 \(\textsf{iP}^{-}\).

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic \(\mathbf{IL}\). We will introduce the logics \(\mathbf{BIL}\) and \(\mathbf{BIL}^{+}\) in the propositional language with a modal operator \(\square\) and a binary operator \(\Rightarrow\) such that \(\mathbf{BIL}\subseteq\mathbf{BIL}^{+}\subseteq\mathbf{IL}\). The logic \(\mathbf{BIL}\) is generated by the relational structures \(\left \), called basic frames, where \(\left \) is a Kripke frame and \(\left \) is a neighborhood frame. We will prove that the logic \(\mathbf{BIL}^{+}\) is generated by the basic frames where the binary relation \(R\) is definable by the neighborhood relation \(N\) and, therefore, the neighborhood semantics is suitable to study the logic \(\mathbf{BIL}^{+}\) and its extensions. We shall also study some axiomatic extensions of \(\mathsf{\mathbf{BIL}}\) and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic BIL+ and some of its extensions are complete respect with the class of neighborhood frames.

In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions.

A subresiduated lattice is a pair, where A is a bounded distributive lattice, D is a bounded sublattice of A and for every there exists the maximum of the set, which is denoted by. This pair can be regarded as an algebra of type (2, 2, 2, 0, 0), where. The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by, whose members satisfy the equation. Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition for every are satisfied, we also study the subvariety of generated by the class whose members satisfy that for every.

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie. In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊. We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in Celani and Montangie. We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$ IntK ◊. We also determine the congruences of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$ H ◊ ∨ -algebras.

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [ 10 ]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces $${\langle X, \leq, T, R \rangle}$$ where $${\langle X, \leq, T \rangle}$$ is a WH -space [ 6 ], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces.

In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras.

The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2]

In this note we introduce the variety $${{\mathcal C}{\mathcal D}{\mathcal M}_\square}$$ of classical modal De Morgan algebras as a generalization of the variety $${{{\mathcal T}{\mathcal M}{\mathcal A}}}$$ of Tetravalent Modal algebras studied in [ 11 ]. We show that the variety $${{\mathcal V}_0}$$ defined by H. P. Sankappanavar in [ 13 ], and the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [ 5 ], are examples of classical modal De Morgan algebras. We give a representation theory, and we study the regular filters, i.e., lattice filters closed under an implication operation. Finally we prove that the variety $${{{\mathcal T}{\mathcal M}{\mathcal A}}}$$ has the Amalgamation Property and the Superamalgamation Property.

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the \-subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures \, where X is a non-empty set and \ is non-empty family of subsets of X such that \. This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator r preserving finite meets which also satisfies the equation?? b V, for all a,b? A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces where is a WH space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH- algebras with successor and the WTf- algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH-spaces.

In this paper we introduce Hilbert algebras with Hilbert–Galois connections (HilGC-algebras) and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting spaces endowed with a binary relation with certain special continuous maps.

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

In this paper we consider the notion of subordination on distributive lattices, equivalent to that of quasi-modal operator for distributive lattices introduced by CastroCastro, J. and Celani in 2004Celani, S.. We provide topological dualities for categories of distributive lattices withDistributive lattices with operators a subordination and then for some categories of distributive lattices with two subordinations, structures that we name bi-subordination lattices. We investigate three classes of bi-subordination lattices. In particular that of positive bi-subordination lattices.

Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined our main objective is to prove completeness with respect to its characteristic class of monotonic frames.

In this work we will study Tarski algebras endowed with a subordination, called subordination Tarski algebras. We will define the notion of round filters, and we will study the class of irreducible round filters and the maximal round filters, called ends. We will prove that the poset of all round filters is a lattice isomorphic to the lattice of the congruences that are compatible with the subordination. We will prove that every end is an irreducible round filter, and that in a topological subordination Tarski algebra A, every irreducible round filter is an end iff A is a monadic subordination Tarski algebra. As corollary of this result we have that the variety of monadic Tarski algebra can be characterised as the topological Tarski algebras where every irreducible open filter is a maximal open filter.

The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions of the minimal system of normal positive modal logic