Quantum MV algebras

We introduce the notion of quantum MV algebra (QMV algebra) as a generalization of MV algebras and we show that the class of all effects of any Hilbert space gives rise to an example of such a structure. We investigate some properties of QMV algebras and we prove that QMV algebras represent non-idempotent extensions of orthomodular lattices.I should like to thank Prof. M.L. Dalla Chiara and Dr. P. Minari for many interesting comments and remarks. Daniele Mundici     

Łukasiewicz-Moisil Relation Algebras

We introduce Łukasiewicz-Moisil relation algebras, obtained by considering a relational dimension over Łukasiewicz-Moisil algebras. We prove some arithmetical properties, provide a characterization in terms of complex algebras, study the connection with relational Post algebras and characterize the simple structures and the matrix relation algebras.     

Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of ?ukasiewicz n-valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n-valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal ?ukasiewicz n-valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal algebras to the n-valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics.     

Categoricity Spectra for Polymodal Algebras

We investigate effective categoricity for polymodal algebras (i.e., Boolean algebras with distinguished modalities). We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.     

Concerning axiomatizability of the quasivariety generated by a finite Heyting or topological Boolean algebra

In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve problem 4 asked in Rautenberg [10].     

Quasivarieties and Congruence Permutability of ?ukasiewicz Implication Algebras

In this paper we study some questions concerning ?ukasiewicz implication algebras. In particular, we show that every subquasivariety of ?ukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite ?ukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.     

Ordered domain algebras

We give a finite axiomatisation to representable ordered domain algebras and show that finite algebras are representable on finite bases.     

Grishin Algebras and Cover Systems for Classical Bilinear Logic

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (??propositions??) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.     

Some Results on Amalgamation in Algebraic Logic

We solve some open questions on the amalgamation porperty invarious algebras of logic (like cylindric algebras). Some ofthese problems first appeared in Pigozzi [3].     

Discrete Dualities for Double Stone Algebras

We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.     

Averaging the truth-value in Łukasiewicz logic

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.     

Partial Boolean algebras in a broader sense

The article deals with compatible families of Boolean algebras. We define the notion of a partial Boolean algebra in a broader sense (PBA(bs)) and then we show that there is a mutual correspondence between PBA(bs) and compatible families of Boolean algebras (Theorem (1.8)). We examine in detail the interdependence between PBA(bs) and the following classes: partial Boolean algebras in the sense of Kochen and Specker (§ 2), ortholattices (§ 3, § 5), and orthomodular posets (§ 4), respectively.     

Type Logics and Pregroups

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller     

Varieties of Monadic Heyting Algebras. Part I

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].     

Varieties of Pseudo-Interior Algebras

The notion of a pseudo-interior algebra was introduced by Blok and Pigozzi in [BPIV]. We continue here our studies begun in [BK]. As a consequence of the representation theorem for pseudo-interior algebras given in [BK] we prove that the variety of all pseudo-interior algebras is generated by its finite members. This result together with Jónsson's Theorem for congruence distributive varieties provides a useful technique in the study of the lattice of varieties of pseudo-interior algebras.     

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.     

A Generalization of Monadic n-Valued Łukasiewicz Algebras

Studia Logica - $${{\mathcal {M}} L}^{m}_n$$ of monadic m-generalized ?ukasiewicz algebras of order n (or $$M L^{m}_n$$ -algebras), namely a generalization of monadic n-valued ?ukasiewicz...