150 research outputs found
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
Colimits of Heyting Algebras through Esakia Duality
In this note we generalize the construction, due to Ghilardi, of the free
Heyting algebra generated by a finite distributive lattice, to the case of
arbitrary distributive lattices. Categorically, this provides an explicit
construction of a left adjoint to the inclusion of Heyting algebras in the
category of distributive lattices This is shown to have several applications,
both old and new, in the study of Heyting algebras: (1) it allows a more
concrete description of colimits of Heyting algebras, as well as, via duality
theory, limits of Esakia spaces, by knowing their description over distributive
lattices and Priestley spaces; (2) It allows a direct proof of the amalgamation
property for Heyting algebras, and of related facts; (3) it allows a proof of
the fact that the category of Heyting algebras is co-distributive. We also
study some generalizations and variations of this construction to different
settings. First, we analyse some subvarieties of Heyting algebras -- such as
Boolean algebras, and algebras, and show how the
construction can be adapted to this setting. Second, we study the relationship
between the category of image-finite posets with p-morphisms and the category
of posets with monotone maps, showing that a variation of the above ideas
provides us with an appropriate general idea.Comment: 27 page
Valuations in Nilpotent Minimum Logic
The Euler characteristic can be defined as a special kind of valuation on
finite distributive lattices. This work begins with some brief consideration on
the role of the Euler characteristic on NM algebras, the algebraic counterpart
of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified
version of the Euler characteristic we call idempotent Euler characteristic. We
show that the new valuation encodes information about the formul{\ae} in NM
propositional logic
Algebraic structure and characterization of adjoint triples
Implications pairs, adjoint pairs and adjoint triples provide general residuated structures considered in different mathematical theories. In this paper, we carry out a deep study on the operators involved in these structures, showing how they are characterized by means of the irreducible elements of a complete lattice. Moreover, the structure of each class of these operators will be analyzed. As a consequence, the use of these operators in real problems will be more tractable, fostering their consideration as basic and useful operators for providing, for instance, preferences among attributes and objects in a given database.Partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects TIN2016-76653-P and PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124
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