1,797 research outputs found
Boolean Lifting Property for Residuated Lattices
In this paper we define the Boolean Lifting Property (BLP) for residuated
lattices to be the property that all Boolean elements can be lifted modulo
every filter, and study residuated lattices with BLP. Boolean algebras, chains,
local and hyperarchimedean residuated lattices have BLP. BLP behaves
interestingly in direct products and involutive residuated lattices, and it is
closely related to arithmetic properties involving Boolean elements, nilpotent
elements and elements of the radical. When BLP is present, strong
representation theorems for semilocal and maximal residuated lattices hold.Comment: 28 page
Taking Prime, Maximal and Two--class Congruences Through Morphisms
In this paper we study prime, maximal and two--class congruences from the
point of view of the relationships between them in various kinds of universal
algebras, as well as their direct and inverse images through morphisms. This
research has also produced a set of interesting results concerning the prime
and the maximal congruences of several kinds of lattices.Comment: 22 page
Congruence Boolean Lifting Property
We introduce and study the Congruence Boolean Lifting Property (CBLP) for
congruence--distributive universal algebras, as well as a property related to
CBLP, which we have called . CBLP extends the so--called Boolean
Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated
lattices, but differs from the BLP when particularized to bounded distributive
lattices. Important classes of universal algebras, such as discriminator
varieties, fulfill the CBLP. The main results of the present paper include a
characterization theorem for congruence--distributive algebras with CBLP and a
structure theorem for semilocal arithmetical algebras with CBLP. When we
particularize the CBLP to the class of residuated lattices and to that of
bounded distributive lattices and we study its relation to other Boolean
Lifting Properties for these algebras, interesting results concerning the image
of the reticulation functor between these classes are revealed.Comment: 30 page
An Algebraic Glimpse at Bunched Implications and Separation Logic
We overview the logic of Bunched Implications (BI) and Separation Logic (SL)
from a perspective inspired by Hiroakira Ono's algebraic approach to
substructural logics. We propose generalized BI algebras (GBI-algebras) as a
common framework for algebras arising via "declarative resource reading",
intuitionistic generalizations of relation algebras and arrow logics and the
distributive Lambek calculus with intuitionistic implication. Apart from
existing models of BI (in particular, heap models and effect algebras), we also
cover models arising from weakening relations, formal languages or more
fine-grained treatment of labelled trees and semistructured data. After briefly
discussing the lattice of subvarieties of GBI, we present a suitable duality
for GBI along the lines of Esakia and Priestley and an algebraic proof of cut
elimination in the setting of residuated frames of Galatos and Jipsen. We also
show how the algebraic approach allows generic results on decidability, both
positive and negative ones. In the final part of the paper, we gently introduce
the substructural audience to some theory behind state-of-art tools,
culminating with an algebraic and proof-theoretic presentation of
(bi-)abduction.Comment: The accepted 2018 final version has been thoroughly rewritten and
improved. To appear in "Outstanding Contributions: Hiroakira Ono on
Residuated Lattices and Substructural Logics
Conceptual Collectives
The notions of formal contexts and concept lattices, although introduced by
Wille only ten years ago, already have proven to be of great utility in various
applications such as data analysis and knowledge representation. In this paper
we give arguments that Wille's original notion of formal context, although
quite appealing in its simplicity, now should be replaced by a more semantic
notion. This new notion of formal context entails a modified approach to
concept construction. We base our arguments for these new versions of formal
context and concept construction upon Wille's philosophical attitude with
reference to the intensional aspect of concepts. We give a brief development of
the relational theory of formal contexts and concept construction,
demonstrating the equivalence of "concept-lattice construction" of Wille with
the well-known "completion by cuts" of MacNeille. Generalization and
abstraction of these formal contexts offers a powerful approach to knowledge
representation.Comment: 30 pages, 11 tables, 4 figures, technical report 199
Some kinds of -fuzzy filters of -algebras
The concepts of -fuzzy
(implicative, positive implicative and fantastic) filters of -algebras are
introduced and some related properties are investigated. Some characterizations
of these generalized fuzzy filters are derived. In particular, we describe the
relationships among ordinary fuzzy (implicative, positive implicative and
fantastic) filters, (\in,\ivq)-fuzzy (implicative, positive implicative and
fantastic) filters and -fuzzy (implicative, positive implicative and fantastic) filters
of -algebras. Finally, we prove that a fuzzy set on a -algebra
is an -fuzzy implicative
filter of if and only if it is both -fuzzy positive implicative filter and an
-fuzzy fantastic filter
On the investigations of Ivan Prodanov in the theory of abstract spectra
The results of Iv. Prodanov on abstract spectra and separative algebras were
announced in the journal "Trudy Mat. Inst. Steklova", 154, 1983, 200--208, but
their proofs were never written by him in the form of a manuscript, preprint or
paper. Since the untimely death of Ivan Prodanov withheld him from preparing
the full version of this paper and since, in our opinion, it contains
interesting and important results, we undertook the task of writing a full
version of it and thus making the results from it known to the mathematical
community. So, the aim of this paper is to supply with proofs the results of
Ivan Prodanov announced in the cited above paper, but we added also a small
amount of new results. The full responsibility for the correctness of the
proofs of the assertions presented below in this work is taken by us; just for
this reason our names appear as authors of the present paper.Comment: 51 page
The relationship between two commutators
We clarify the relationship between the linear commutator and the ordinary
commutator by showing that in any variety satisfying a nontrivial idempotent
Mal'cev condition the linear commutator is definable in terms of the
centralizer relation. We derive from this that abelian algebras are
quasi-affine in such varieties. We refine this by showing that if A is an
abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails
to hold in the variety of semilattices, then A is affine
The origin of spacetime topology and generalizations of quantum field theory
The research effort reported in this paper is directed, in a broad sense,
towards understanding the small-scale structure of spacetime. The fundamental
question that guides our discussion is ``what is the physical content of
spacetime topology?" In classical physics, if spacetime, , has
sufficiently regular topology, and if sufficiently many fields exist to allow
us to observe all continuous functions on , then this collection of
continuous functions uniquely determines both the set of points and the
topology on it. To explore the small-scale structure of spacetime, we
are led to consider the physical fields (the observables) not as classical
(continuous functions) but as quantum operators, and the fundamental observable
as not the collection of all continuous functions but the local algebra of
quantum field operators. In pursuing our approach further, we develop a number
of generalizations of quantum field theory through which it becomes possible to
talk about quantum fields defined on arbitrary topological spaces. Our ultimate
generalization dispenses with the fixed background topological space altogether
and proposes that the fundamental observable should be taken as a lattice (or
more specifically a ``frame," in the sense of set theory) of closed subalgebras
of an abstract algebra. Our discussion concludes with the definition
and some elementaryComment: 23 pages, UCSBTH-92-4
Join-completions of ordered algebras
We present a systematic study of join-extensions and join-completions of
ordered algebras, which naturally leads to a refined and simplified treatment
of fundamental results and constructions in the theory of ordered structures
ranging from properties of the Dedekind-MacNeille completion to the proof of
the finite embeddability property for a number of varieties of ordered
algebras
- …