55 research outputs found
Fuzzy closure systems: Motivation, definition and properties
The aim of this paper is to extend closure systems from being crisp sets with certain
fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough
discussion on the different alternatives that could be taken to define the desired fuzzy
closure systems. These plausible alternatives are discarded if they are proven impossible
to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy
closure system is established and a one-to-one relation with closure operators is proved.The aim of this paper is to extend closure systems from being crisp sets with certain
fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough
discussion on the different alternatives that could be taken to define the desired fuzzy
closure systems. These plausible alternatives are discarded if they are proven impossible
to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy
closure system is established and a one-to-one relation with closure operators is proved. Funding for open access charge: Universidad de Málaga / CBU
On Contextuality and Unsharp Quantum Logic
In this paper we provide a preliminary investigation of subclasses of bounded
posets with antitone involution which are "pastings" of their maximal Kleene
sub-lattices. Specifically, we introduce super-paraorthomodular lattices,
namely paraothomodular lattices whose order determines, and it is fully
determined by, the order of their maximal Kleene sub-algebras. It will turn out
that the (spectral) paraorthomodular lattice of effects over a separable
Hilbert space can be considered as a prominent example of such. Therefore, it
arguably provides an algebraic/order theoretical rendering of complementarity
phenomena between unsharp observables. A number of examples, properties and
characterization theorems for structures we deal with will be outlined. For
example, we prove a forbidden configuration theorem and we investigate the
notion of commutativity for modular pseudo-Kleene lattices, examples of which
are (spectral) paraorthomodular lattices of effects over finite-dimensional
Hilbert spaces. Finally, we show that structures introduced in this paper yield
paraconsistent partial referential matrices, the latter being generalizations
of J. Czelakowski's partial referential matrices. As a consequence, a link
between some classes of posets with antitone involution and algebras of partial
"unsharp" propositions is established
On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics
The aim of this article is to explore the class of intermediate logics between the truth-preserving Łukasiewicz logic Ł and its degree-preserving companion Ł≤. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in Ł≤ and derivable in L and we characterize the corresponding intermediate logics. From a semantical point of view, we first consider the family of logics characterized by matrices defined by lattice filters in [0,1], but we show there are intermediate logics falling outside this family. Finally, we study the case of finite-valued Lukasiewicz logics where we axiomatize a large family of intermediate logics defined by families of matrices (A,F) such that A is a finite MV-algebra and F is a lattice filter.The authors have been partially supported by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Coniglio was also supported by FAPESP (Thematic Project LogCons 2010/51038-0), and by a research grant from CNPq
(PQ 308524/2014-4). Esteva and Godo also acknowledge partial support by the MINECO project TIN2012-39348-C02-01.Peer Reviewe
Compatible Quantum Theory
Formulations of quantum mechanics can be characterized as realistic,
operationalist, or a combination of the two. In this paper a realistic theory
is defined as describing a closed system entirely by means of entities and
concepts pertaining to the system. An operationalist theory, on the other hand,
requires in addition entities external to the system. A realistic formulation
comprises an ontology, the set of (mathematical) entities that describe the
system, and assertions, the set of correct statements (predictions) the theory
makes about the objects in the ontology. Classical mechanics is the prime
example of a realistic physical theory. The present realistic formulation of
the histories approach originally introduced by Griffiths, which we call
'Compatible Quantum Theory (CQT)', consists of a 'microscopic' part (MIQM),
which applies to a closed quantum system of any size, and a 'macroscopic' part
(MAQM), which requires the participation of a large (ideally, an infinite)
system. The first (MIQM) can be fully formulated based solely on the assumption
of a Hilbert space ontology and the noncontextuality of probability values,
relying in an essential way on Gleason's theorem and on an application to
dynamics due in large part to Nistico. The microscopic theory does not,
however, possess a unique corpus of assertions, but rather a multiplicity of
contextual truths ('c-truths'), each one associated with a different framework.
This circumstance leads us to consider the microscopic theory to be physically
indeterminate and therefore incomplete, though logically coherent. The
completion of the theory requires a macroscopic mechanism for selecting a
physical framework, which is part of the macroscopic theory (MAQM). Detailed
definitions and proofs are presented in the appendice
Consensus theories: an oriented survey
This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity
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