82 research outputs found
Primeness in Quantales
In this paper we propose a new concept of primeness in quantales. It is proved that this concept coincide with classical definition in commutative quantales, but no longer valid in the noncommutative setting. Also, the notions of strong and uniform strong primeness are investigated
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
On Pythagorean fuzzy ideals of a classical ring
The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals
The quantum chiral Minkowski and conformal superspaces
We give a quantum deformation of the chiral super Minkowski space in four
dimensions as the big cell inside a quantum super Grassmannian. The
quantization is performed in such way that the actions of the Poincar\'e and
conformal quantum supergroups on the quantum Minkowski and quantum conformal
superspaces are presented.Comment: 54 page
Algebraic Structures using Natural Class of Intervals
This book has eleven chapters. Chapter one describes all types of natural
class of intervals and the arithmetic operations on them. Chapter two
introduces the semigroup of natural class of intervals using R or Zn and study
the properties associated with them. Chapter three studies the notion of rings
constructed using the natural class of intervals. Matrix theory using the
special class of intervals is analyzed in chapter four of this book. Chapter
five deals with polynomials using interval coefficients. New types of rings of
natural intervals are introduced and studied in chapter six. The notion of
vector space using natural class of intervals is built in chapter seven. In
chapter eight fuzzy natural class of intervals are introduced and algebraic
structures on them is built and described. Algebraic structures using natural
class of neutrosophic intervals are developed in chapter nine.Chapter ten
suggests some possible applications. The final chapter proposes over 200
problems of which some are at research level and some difficult and others are
simple.Comment: 170 pages; Published by The Educational Publisher Inc in 201
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