288 research outputs found
Dicomplemented Lattices: A Contextual Generalization of Boolean Algebras
Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen.The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a "negation" on formal concepts, "concept algebras" are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on "Contextual Logic", a Formal Concept Analysis approach, based on concepts as units of thought
Gautama and Almost Gautama Algebras and their associated logics
Recently, Gautama algebras were defined and investigated as a common generalization of the variety of regular double Stone algebras and the variety of regular Kleene Stone algebras, both of which are, in turn, generalizations of Boolean algebras. Those algebras were named in honor and memory of the two founders of Indian Logic--{\bf Akshapada Gautama} and {\bf Medhatithi Gautama}. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called ``Almost Gautama algebras (, for short).'' More precisely, we give an explicit description of subdirectly irreducible Almost Gautama algebras. As consequences, explicit description of the lattice of subvarieties of and the equational bases for all its subvarieties are given. It is also shown that the variety is a discriminator variety. Next, we consider logicizing ; but the variety lacks an implication operation. We, therefore, introduce another variety of algebras called ``Almost Gautama Heyting algebras'' (, for short) and show that the variety %of Almost Heyting algebras is term-equivalent to that of . Next, a propositional logic, called (or ), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety , via as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic , corresponding to all the subvarieties of are given. They include the axiomatic extensions , and of the logic corresponding to the varieties , , and (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of has the Disjunction Property. Finally, We revisit the classical logic with strong negation and classical Nelson algebras introduced by Vakarelov in 1977 and improve his results by showing that is algebraizable with as its algebraic semantics and that the logics , , 3-valued \L ukasivicz logic and the classical logic with strong negation are all equivalent.Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Sankappanavar, Hanamantagouda P.. State University of New York. Department of Mathematics ; Estados Unido
Checking Admissibility Using Natural Dualities
This paper presents a new method for obtaining small algebras to check the
admissibility-equivalently, validity in free algebras-of quasi-identities in a
finitely generated quasivariety. Unlike a previous algebraic approach of
Metcalfe and Rothlisberger that is feasible only when the relevant free algebra
is not too large, this method exploits natural dualities for quasivarieties to
work with structures of smaller cardinality and surjective rather than
injective morphisms. A number of case studies are described here that could not
be be solved using the algebraic approach, including (quasi)varieties of
MS-algebras, double Stone algebras, and involutive Stone algebras
Contributions to the theory of Ockham algebras
In the first part of this thesis we consider particular ordered sets (connected and of small height) and determine the cardinality of the corresponding dual MS - algebra and of its set of fixed points.
The remainder of the thesis is devoted to a study of congruences of Ockham algebras and a generalised variety K of Ockham algebras that contains all of the Berman varieties K[sub]p,[sub]q. In particular we consider the congruences [sub]i(i = 1, 2,...) defined on an Ockham algebra (L; f) by
(x, y) ∊ [sub]i ⇔ fⁱ(x)= fⁱ(y)
and show that (L; f) ∊ K is subdirectly irreducible if and only if the lattice of congruences of L reduces to the chain
= ₀ ≤ ₁≤ ₂≤ … ≤<
Where = ⌵ [sub]i≥0i. Finally we obtain a characterisation of the finite simple Ockham algebras
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