611 research outputs found
Information completeness in Nelson algebras of rough sets induced by quasiorders
In this paper, we give an algebraic completeness theorem for constructive
logic with strong negation in terms of finite rough set-based Nelson algebras
determined by quasiorders. We show how for a quasiorder , its rough
set-based Nelson algebra can be obtained by applying the well-known
construction by Sendlewski. We prove that if the set of all -closed
elements, which may be viewed as the set of completely defined objects, is
cofinal, then the rough set-based Nelson algebra determined by a quasiorder
forms an effective lattice, that is, an algebraic model of the logic ,
which is characterised by a modal operator grasping the notion of "to be
classically valid". We present a necessary and sufficient condition under which
a Nelson algebra is isomorphic to a rough set-based effective lattice
determined by a quasiorder.Comment: 15 page
Web spaces and worldwide web spaces: Topological aspects of domain theory
Web spaces, wide web spaces and worldwide web spaces (alias C-spaces) provide useful generalizations of continuous domains. We present new characterizations of such spaces and their patch spaces, obtained by joining the original topology with a second topology having the dual specialization order; these patch spaces possess good convexity and separation properties and determine the original web spaces. The category of C-spaces is concretely isomorphic to the category of fan spaces; these are certain quasi-ordered spaces having neighborhood bases of fans, where a fan is obtained by deleting a finite number of principal dual ideals from a principal dual ideal. Our approach has useful consequences for domain theory, because the T0 web spaces are exactly the generalized Scott spaces associated with locally approximating ideal extensions, and the T0 C-spaces are exactly the generalized Scott spaces associated with globally approximating and interpolating ideal extensions. The characterization of continuous lattices as meet-continuous lattices with T2 Lawson topology and the Fundamental Theorem of Compact Semilattices are extended to non-complete posets. Finally, cardinal invariants like density and weight of the involved objects are investigated. © 2019 Marcel Erné. All rights reserved
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Tensor products of subspace lattices and rank one density
We show that, if is a subspace lattice with the property that the rank
one subspace of its operator algebra is weak* dense, is a commutative
subspace lattice and is the lattice of all projections on a separable
infinite dimensional Hilbert space, then the lattice is
reflexive. If is moreover an atomic Boolean subspace lattice while is
any subspace lattice, we provide a concrete lattice theoretic description of
in terms of projection valued functions defined on the set of
atoms of . As a consequence, we show that the Lattice Tensor Product Formula
holds for \Alg M and any other reflexive operator algebra and give several
further corollaries of these results.Comment: 15 page
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
A theory of quantale-enriched dcpos and their topologization
There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral
and commutative, then Q-valued domains are Q-enriched, and every Q-enriched domain is sober in its Scott Q-valued topology,
where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and
Systems (2022)). In this paper, we provide a framework for the development of Q-enriched dcpos and Q-enriched domains in the
general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right
subdistributive quasi-magmas on Q in the sense of the category Cat(Q). It is important to point out that our quasi-magmas on Q
are in tune with the «intersection axiom» of Q-enriched topologies. When Q is involutive, each Q-enriched domain becomes sober
in its Q-enriched Scott topology. This paper also offers a perspective to apply Q-enriched dcpos to quantale computatio
- …