25 research outputs found
Reflective Full Subcategories of the Category of L
This paper focuses on the relationship between L-posets and complete L-lattices from the categorical view. By considering a special class of fuzzy closure operators, we prove that the category of complete L-lattices is a reflective full subcategory of the category of L-posets with appropriate morphisms. Moreover, we characterize the Dedekind-MacNeille completions of L-posets and provide an equivalent description for them
Quantitative Concept Analysis
Formal Concept Analysis (FCA) begins from a context, given as a binary
relation between some objects and some attributes, and derives a lattice of
concepts, where each concept is given as a set of objects and a set of
attributes, such that the first set consists of all objects that satisfy all
attributes in the second, and vice versa. Many applications, though, provide
contexts with quantitative information, telling not just whether an object
satisfies an attribute, but also quantifying this satisfaction. Contexts in
this form arise as rating matrices in recommender systems, as occurrence
matrices in text analysis, as pixel intensity matrices in digital image
processing, etc. Such applications have attracted a lot of attention, and
several numeric extensions of FCA have been proposed. We propose the framework
of proximity sets (proxets), which subsume partially ordered sets (posets) as
well as metric spaces. One feature of this approach is that it extracts from
quantified contexts quantified concepts, and thus allows full use of the
available information. Another feature is that the categorical approach allows
analyzing any universal properties that the classical FCA and the new versions
may have, and thus provides structural guidance for aligning and combining the
approaches.Comment: 16 pages, 3 figures, ICFCA 201
Residuated structures and orthomodular lattices
The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., ℓ-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated ℓ-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated ℓ-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated ℓ-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices
Archimedean atomic lattice effect algebras in which all sharp elements are central
summary:We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it
Quantaloidal Completions of Order-enriched Categories and Their Applications
By introducing the concept of quantaloidal completions for an order-enriched
category, relationships between the category of quantaloids and the category of
order-enriched categories are studied. It is proved that quantaloidal
completions for an order-enriched category can be fully characterized as
compatible quotients of the power-set completion. As applications, we show that
a special type of injective hull of an order-enriched category is the MacNeille
completion; the free quantaloid over an order-enriched category is the Down-set
completion
Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements
We study Archimedean atomic lattice effect algebras whose set of sharp
elements is a complete lattice. We show properties of centers, compatibility
centers and central atoms of such lattice effect algebras. Moreover, we prove
that if such effect algebra is separable and modular then there exists a
faithful state on . Further, if an atomic lattice effect algebra is densely
embeddable into a complete lattice effect algebra and the
compatiblity center of is not a Boolean algebra then there exists an
-continuous subadditive state on
Pseudo-Kleene algebras determined by rough sets
We study the pseudo-Kleene algebras of the Dedekind-MacNeille completion of
the ordered set of rough set determined by a reflexive relation. We
characterize the cases when PBZ and PBZ*-lattices can be defined on these
pseudo-Kleene algebras.Comment: 24 pages, minor update to the initial versio
Preservation theorems for algebraic and relational models of logic
A thesis submitted to the School of Computer Science,
Faculty of Science,
University of the Witwatersrand, Johannesburg
in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013In this thesis a number of different constructions on ordered algebraic structures
are studied. In particular, two types of constructions are considered: completions
and finite embeddability property constructions.
A main theme of this thesis is to determine, for each construction under
consideration, whether or not a class of ordered algebraic structures is closed
under the construction. Another main focus of this thesis is, for a particular
construction, to give a syntactical description of properties preserved by the
construction. A property is said to be preserved by a construction if, whenever
an ordered algebraic structure satisfies it, then the structure obtained through
the construction also satisfies the property.
The first four constructions investigated in this thesis are types of completions.
A completion of an ordered algebraic structure consists of a completely
lattice ordered algebraic structure and an embedding that embeds the former
into the latter. Firstly, different types of filters (dually, ideals) of partially ordered
sets are investigated. These are then used to form the filter (dually, ideal)
completions of partially ordered sets. The other completions of ordered algebraic
structures studied here include the MacNeille completion, the canonical
extension (also called the completion with respect to a polarization) and finally
a prime filter completion.
A class of algebras has the finite embeddability property if every finite partial
subalgebra of some algebra in the class can be embedded into some finite
algebra in the class. Firstly, two constructions that establish the finite embeddability
property for residuated ordered structures are investigated. Both of
these constructions are based on completion constructions: the first on the Mac-
Neille completion and the second on the canonical extension. Finally, algebraic
filtrations on modal algebras are considered and a duality between algebraic and
relational versions of filtrations is established
Archimedean atomic lattice effect algebras with complete lattice of sharp elements
We study Archimedean atomic lattice effect algebras whose set of sharp elements
is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice ef fect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra Eb and the compatibility center of E is not a Boolean algebra then there exists an (o)-continuous subadditive state on E