The Variety of Lattice Effect Algebras Generated by MV-algebras and the Horizontal Sum of Two 3-element Chains

It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and is definable by three simple identities and the problem now is to check if these identities are satisfied by all distributive lattice effect algebras or not. Presented by Daniele Mundici     

Varieties of Three-Valued Heyting Algebras with a Quantifier

This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q ₃ of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q ₃ and we construct the lattice of subvarieties (Q ₃ ) of the variety Q ₃ .     

Distributive lattices with an operator

It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences.I would like to thank my research supervisor Dr. Roberto Cignoli for his helpful suggestions during the preparation of this paper and the referee for calling my attention to Goldblatt's paper [5].     

An Infinitary Extension of Jankov’s Theorem

It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra B, A is embeddable into a quotient algebra of B, if and only if Jankov’s formula χ _A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number κ, we present Jankov’s theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly irreducible complete κ-Heyting algebras and κ-infinitary logic, where a κ-Heyting algebra is a Heyting algebra A with # ≥  κ and κ-infinitary logic is the infinitary logic such that for any set Θ of formulas with # Θ ≥  κ, ∨Θ and ∧Θ are well defined formulas.     

Reductio ad contradictionem: An Algebraic Perspective

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms.     

Deontic action logic, atomic boolean algebras and fault-tolerance

We introduce a deontic action logic and its axiomatization. This logic has some useful properties (soundness, completeness, compactness and decidability), extending the properties usually associated with such logics. Though the propositional version of the logic is quite expressive, we augment it with temporal operators, and we outline an axiomatic system for this more expressive framework. An important characteristic of this deontic action logic is that we use boolean combinators on actions, and, because of finiteness restrictions, the generated boolean algebra is atomic, which is a crucial point in proving the completeness of the axiomatic system. As our main goal is to use this logic for reasoning about fault-tolerant systems, we provide a complete example of a simple application, with an attempt at formalization of some concepts usually associated with fault-tolerance.     

Homogeneous and Universal Dedekind Algebras

A Dedekind algebra is an order pair (B, h) where B is a non-empty set and h is a similarity transformation on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type which occur in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. It is shown that configuration signatures can be used to characterize the homogeneous, universal and homogeneous-universal Dedekind algebras. This characterization is used to prove various results about these subclasses of Dedekind algebras.     

Algebras and Matrices for Annotated Logics

We study the matrices, reduced matrices and algebras associated to the systems SA_T of structural annotated logics. In previous papers, these systems were proven algebraizable in the finitary case and the class of matrices analyzed here was proven to be a matrix semantics for them.We prove that the equivalent algebraic semantics associated with the systems SA_T are proper quasivarieties, we describe the reduced matrices, the subdirectly irreducible algebras and we give a general decomposition theorem. As a consequence we obtain a decision procedure for these logics.     

Subdirectly Irreducible Modal Algebras and Initial Frames

The duality between general frames and modal algebras allows to transfer a problem about the relational (Kripke) semantics into algebraic terms, and conversely. We here deal with the conjecture: the modal algebra A is subdirectly irreducible (s.i.) if and only if the dual frame A* is generated. We show that it is false in general, and that it becomes true under some mild assumptions, which include the finite case and the case of K4. We also prove that a Kripke frame F is generated if and only if the dual algebra F* is s.i. The technical result is that A is s.i. when the set of points which generate the dual frame A* is not of zero measure.     

The First-Order Theories of Dedekind Algebras

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence.     

Optimal natural dualities for varieties of Heyting algebras

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.     

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ?₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.     

Birkhoff-like sheaf representation for varieties of lattice expansions

Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.We wish to thank Lic. Alfredo Guerin and Dr. Daniel Penazzi for helping us with linguistics aspects. We are indebted to the referee for several helpful suggestions. We also wish to thank Professor Mick Adams for providing us with several reprints and useful e-mail information on the subject.Suported by CONICOR and SECyT (UNC).     

Quasivarieties with Definable Relative Principal Subcongruences

For quasivarieties of algebras, we consider the property of having definable relative principal subcongruences, a generalization of the concepts of definable relative principal congruences and definable principal subcongruences. We prove that a quasivariety of algebras with definable relative principal subcongruences has a finite quasiequational basis if and only if the class of its relative (finitely) subdirectly irreducible algebras is strictly elementary. Since a finitely generated relatively congruence-distributive quasivariety has definable relative principal subcongruences, we get a new proof of the result due to D. Pigozzi: a finitely generated relatively congruence-distributive quasivariety has a finite quasi-equational basis. Presented by W. Dziobiak     

Galois structures

This paper is a continuation of investigations on Galois connections from [1], [3], [10]. It is a continuation of [2]. We have shown many results that link properties of a given closure space with that of the dual space. For example: for every -disjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the -prime sets in X (Theorem 2). Moreover the characterizations of the satisfiability relation for classical logic are shown. Roughly speaking our main result here is the following: a satisfiability relation in a logic L with, a countable language is a fragment of the classical one iff the compactness theorem for L holds (Theorems 3–8).     

Negation in the Context of Gaggle Theory

We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic K_i. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. K_u is the minimal logic that has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic K_u. Finally, we unite the two basic logics K_i and K_u together to get a negative modal logic K_-, which is dual to the positive modal logic K₊ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations.     

Semi-DeMorgan algebras

Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of partial diagrams is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest variety of semi-DeMorgan algebras with the congruence extension property.Semi-deMorgan algebras include demi-p-lattices, the topic of H. Gaitan's contribution to this special edition. D. Hobby's results were obtained independently.