Dualities for Plonka sums

Plonka sums consist of an algebraic construction similar, in some sense to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Plonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Plonka sum representation in terms of dualisable algebras.Comment: 12 pages; the paper was awarded with the "SILFS Logic Prize" and is appearing on "Logica Universalis

Logics of left variable inclusion and Płonka sums of matrices

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic ⊢. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic ⊢ is related to the construction of Płonka sums of the matrix models of ⊢. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy

Approximation in Databases

One source of partial information in databases is the need to combine information from several databases. Even if each database is complete for some world , the combined databases will not be, and answers to queries against such combined databases can only be approximated. In this paper we describe various situations in which a precise answer cannot be obtained for a query asked against multiple databases. Based on an analysis of these situations, we propose a classification of constructs that can be used to model approximations. One of the main goals is to show that most of these models of approximations possess universality properties. The main motivation for doing this is applying the data-oriented approach, which turns universality properties into syntax, to obtain languages for approximations. We show that the languages arising from the universality properties have a number of limitations. In an attempt to overcome those limitations, we explain how all the languages can be embedded into a language for conjunctive and disjunctive sets from [21], and demonstrate its usefulness in querying independent databases

An algebraic study of logics of variable inclusion and analytic containment

This thesis focuses on a wide family of logics whose common feature is to admit a syntactic definition based on specific variable inclusion principles. This family has been divided into three main components: logics of left variable inclusion, containment logics, and the logic of demodalised analytic implication. We offer a general investigation of such logics within the framework of modern abstract algebraic logic