Search results for "Interior algebra"

The overlap algebra of regular opens

2010

Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, …

Algebras with involution with linear codimension growth

2006

AbstractWe study the ∗-varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite-dimensional algebra. In this setting we give a list of algebras classifying all such ∗-varieties whose sequence of ∗-codimensions is linearly bounded. Moreover, we exhibit a finite list of algebras to be excluded from the ∗-varieties with such property. As a consequence, we find all possible linearly bounded ∗-codimension sequences.

C*-seminorms on partial *-algebras: an overview

2005

On Rough Sets in Topological Boolean Algebras

1994

We have focused on rough sets in topological Boolean algebras. Our main ideas on rough sets are taken from concepts of Pawlak [4] and certain generalizations of his constructions which were offered by Wiweger [7]. One of the most important results of this note is a characterization of the rough sets determined by regular open and regular closed elements.

Sobriety and spatiality in categories of lattice-valued algebras

2012

The paper provides an analogue of the famous equivalence between the categories of sober topological spaces and spatial locales for the framework of (L,M)-fuzzy topology of Kubiak and Sostak (and partly to that of Guido). To be more general, we replace locales with localic lattice-valued algebras in the sense of Di Nola and Gerla and use the respective generalized topological setting. As a result, it appears that the shift from crisp algebras to lattice-valued algebras weakens (resp. strengthens) considerably the classical (including the point-set lattice-theoretic setting of Rodabaugh) notion of sobriety (resp. spatiality).

Extended-order algebras as a generalization of posets

2011

Motivated by the recent study of several researchers on extended-order algebras, introduced by C. Guido and P. Toto as a possible common framework for the majority of algebraic structures used in many-valued mathematics, the paper focuses on the properties of homomorphisms of the new structures, considering extended-order algebras as a generalization of partially ordered sets. The manuscript also introduces the notion of extended-relation algebra providing a new framework for developing the theory of rough sets.

On monadic quantale algebras: basic properties and representation theorems

2010

Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new structures.

Rough Set Algebras as Description Domains

2009

Study of the so called knowledge ordering of rough sets was initiated by V.W. Marek and M. Truszczynski at the end of 90-ies. Under this ordering, the rough sets of a fixed approximation space form a domain in which every set ↓ is a Boolean algebra. In the paper, an additional operation inversion on rough set domains is introduced and an abstract axiomatic description of obtained algebras of rough set is given. It is shown that the resulting class of algebras is essentially different from those traditional in rough set theory: it is not definable, for instance, in the class of regular double Stone algebras, and conversely.

Almost polynomial growth: Classifying varieties of graded algebras

2015

Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all sub…