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.     

Subdirectly Irreducible Residuated Semilattices and Positive Universal Classes

CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)^k ≤ (x ⋀ e)^k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k). Dedicated to the memory of Willem Johannes Blok     

Free Q-distributive lattices

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.     

The Scope and Limitations of Algebras: Some Historical and Philosophical Considerations

An important feature of mathematics, both pure and applied, during the nineteenth century was the widening from its common form to a proliferation, where the “objects” studied were not numbers or geometrical magnitudes but operations such as functions and differentiation and integration, abstract ones (as we now call them), linear algebras of vectors, matrices and determinants, and algebras in logic. In this article the author considers several of them, including the contributions of Hermann Grassmann and Benjamin Peirce. A notable feature of these developments was analogising from one algebra to another by adopting some of the same laws, such as associativity, commutativity and distributivity. In the final section we consider the normally secular character of these algebras.     

Congruences on Dynamic Algebras

The lattice Cong of all dynamic congruences on a given dynamic algebra is presented. Whenever is separable with zero we define dynamic ideal on , given rise to the lattice Ide. The notions of kernel of a dynamic congruence andthe congruence generated by a dynamic ideal are introduced todescribe a Galois connection between Cong and Ide. We study conditions under which a dynamic congruence is determined byits kernel.     

Interpolation and Amalgamation; Pushing the Limits. Part I

Continuing work initiated by Jónsson, Daigneault, Pigozzi and others; Maksimova proved that a normal modal logic (with a single unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property (cf. [Mak 91], [Mak 79]). The aim of this paper is to extend the latter result to a large class of logics. We will prove that the characterization can be extended to all algebraizable logics containing Boolean fragment and having a certain kind of local deduction property. We also extend this characterization of the interpolation property to arbitrary logics under the condition that their algebraic counterparts are discriminator varieties. We also extend Maksimova's result to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than 2, too.The problem of extending the above characterization result to no n-normal non-unary modal logics remains open.Related issues of universal algebra and of algebraic logic are discussed, too. In particular we investigate the possibility of extending the characterization of interpolability to arbitrary algebraizable logics.     

Interpolation and Amalgamation; Pushing the Limits. Part II

This is the second part of the paper [Part I] which appeared in the previous issue of this journal.     

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).     

Maximal Subalgebras of MVn-algebras. A Proof of a Conjecture of A. Monteiro

For each integer n ≥ 2, MV_n denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MV_n are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MV_n are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV₃, the mentioned characterization of maximal subalgebras of A can be given in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. Mathematics Subject Classification (2000): 06D30, 06D35, 03G20, 03B50, 08A30. Presented by Daniele Mundici     

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators,and Applications to the Semantics of Non-Classical Logics I

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains.     

Dual scaling of comparison and reference stimuli in multi-dimensional psychological space

Dzhafarov and Colonius (Psychol. Bull. Rev. 6 (1999)239; J. Math. Psychol. 45(2001)670) proposed a theory of Fechnerian scaling of the stimulus space based on the psychometric (discrimination probability) function of a human subject in a same-different comparison task. Here, we investigate a related but different paradigm, namely, referent-probe comparison task, in which the pair of stimuli (x and y) under comparison assumes substantively different psychological status, one serving as a referent and the other as a probe. The duality between a pair of psychometric functions, arising from assigning either x or y to be the fixed reference stimulus and the other to be the varying comparison stimulus, and the 1-to-1 mapping between the two stimulus spaces X and Y under either assignment are analyzed. Following Dzhafarov and Colonius, we investigate two properties characteristic of a referent-probe comparison task, namely, (i) Regular cross-minimality—for the pair of stimulus values involved in referent-probe comparison, each minimizes a discrimination probability function where the other is treated as the fixed reference stimulus; (ii) Nonconstant self-similarity—the value of the discrimination probability function at its minima is a nonconstant function of the reference stimulus value. For the particular form of psychometric functions investigated, it is shown that imposing the condition of regular cross-minimality on the pair of psychometric functions forces a consistent (but otherwise still arbitrary) mapping between X and Y, such that it is independent of the assignment of reference/comparison status to x and to y. The resulting psychometric differentials under both assignments are equal, and take an asymmetric, dualistic form reminiscent of the so-called divergence measure that appeared in the context of differential geometry of the probability manifold with dually flat connections (Differential Geometric Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer, New York, 1985). The pair of divergence functions on X and on Y, respectively, induce a Riemannian metric in the small, with psychometric order (defined in Dzhafarov & Colonius, 1999) equal to 2. The difference between the Finsler-Riemann geometric approach to the stimulus space (Dzhafarov & Colonius, 1999) and this dually-affine Riemannian geometric approach to the dual scaling of the comparison and the reference stimuli is discussed.     

Tableaux and Dual Tableaux: Transformation of Proofs

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. Presented by Wojciech Buszkowski     

A Coalgebraic View of Heyting Duality

We give a coalgebraic view of the restricted Priestley duality between Heyting algebras and Heyting spaces. More precisely, we show that the category of Heyting spaces is isomorphic to a full subcategory of the category of all -coalgebras, based on Boolean spaces, where is the functor which maps a Boolean space to its hyperspace of nonempty closed subsets. As an appendix, we include a proof of the characterization of Heyting spaces and the morphisms between them.     

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B ₂ of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B ₂. The axioms of Schmid [7], [8] for the AC and EC members of B ₂ are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B ₂ and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B ₂, this method transfers the entire discussion into an equivalent dual category X ₂ of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X ₂ prove to be much more readily constructed and analyzed than the corresponding algebras in B ₂.     

A Note on Algebras of Substitutions

We will study the class RSA of -dimensional representable substitution algebras. RSA is a sub-reduct of the class of representable cylindric: algebras, and it was an open problem in Andréka [1] that whether RSA can be finitely axiomatized. We will show, that the answer is positive. More concretely, we will prove, that RSA is a finitely axiomatizable quasi-variety. The generated variety is also described. We note that RSA is the algebraic counterpart of a certain proportional multimodal logic and it is related to a natural fragment of first order logic, as well.     

A QBF-based formalization of abstract argumentation semantics

We introduce a unified logical theory, based on signed theories and Quantified Boolean Formulas (QBFs) that can serve as the basis for representing and computing various argumentation-based decision problems. It is shown that within our framework we are able to model, in a simple and modular way, a wide range of semantics for abstract argumentation theory. This includes complete, grounded, preferred, stable, semi-stable, stage, ideal and eager semantics. Furthermore, our approach is purely logical, making for instance decision problems like skeptical and credulous acceptance of arguments simply a matter of entailment and satisfiability checking. The latter may be verified by off-the-shelf QBF-solvers.