Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality.In part III we discuss applications in logic of the framework developed. Specifically, logics with restricted structural rules give rise to lattices with normal operators (in our sense), such as the Full Lambek algebras (F L-algebras) studied by Ono in [36]. Our Stone-type representation results can be then used to obtain canonical constructions of Kripke frames for such systems, and to prove a duality of algebraic and Kripke semantics for such logics.     

Weakly higher order cylindric algebras and finite axiomatization of the representables

We show that the variety of n-dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski’s quasi-projective relation algebras. Supported by the Hungarian National Foundation for Scientific Research grant T73601.     

Quasi-varieties: A special access

Quasi-equational logic concerns with a completeness theorem, i. e. a list of general syntactical rules such that, being given a set of graded quasi-equations Q, the closure Cl Q = Qeq Fun Q can be derived from by the given rules. Those rules do exist, because our consideration could be embedded into the logic of first order language. But, we look for special (quasi-equational) rules. Suitable rules were already established for the (non-functorial) case of partial algebras in Definition 3.1.2 of [27], p. 108, and [28], p. 102. (For the case of total algebras, see [35].) So, one has to translate these rules to the (functorial) language of partial theories .Surprisingly enough, partial theories can be replaced up to isomorphisms by partial Dale monoids (cf. Section 3), which, in the total case are ordinary monoids.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

On Łukasiewicz's Four-Valued Modal Logic

ukasiewicz's four-valued modal logic is surveyed and analyzed, together with ukasiewicz's motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz's own texts, and related literature.     

Weakly Associative Relation Algebras with Polyadic Composition Operations

In this paper we introduced various classes of weakly associative relation algebras with polyadic composition operations. Among them is the class RWA of representable weakly associative relation algebras with polyadic composition operations. Algebras of this class are relativized representable relation algebras augmented with an infinite set of operations of increasing arity which are generalizations of the binary relative composition. We show that RWA is a canonical variety whose equational theory is decidable.     

Linearization of the BCK-logic

In the paper we obtain a new characterization of the BCK-algebras which are subdirect product of BCK-chains. We give an axiomatic algebraizable extension of the BCK-calculus, by means of a recursively enumerable set of axioms, such that its equivalent algebraic semantics is definitionally equivalent to the quasivariety of BCK-algebras generated by the BCK-chains. We propose the concept of "linearization of a system" and we give some examples.     

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.