scholarly journals An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices

ISRN Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Rasouli ◽  
B. Davvaz
Keyword(s):  
Algebraic Structure ◽  
Cartesian Product ◽  
Residuated Lattices ◽  
Soft Sets ◽  
Basic Properties ◽  
The Family

We introduce the notion of soft filters in residuated lattices and investigate their basic properties. We investigate relations between soft residuated lattices and soft filter residuated lattices. The restricted and extended intersection (union), ∨ and ∧-intersection, cartesian product, and restricted and extended difference of the family of soft filters residuated lattices are established. Also, we consider the set of all soft sets over a universe set U and the set of parameters P with respect to U, SoftP(U), and we study its structure.

scholarly journals Convexity of sets in metric Abelian groups

2020 ◽  
Vol 32 (6) ◽  
pp. 1477-1486 ◽  
Author(s):  
Włodzimierz Fechner ◽  
Zsolt Páles
Keyword(s):  
Abelian Group ◽  
Structural Properties ◽  
Spectral Radius ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Invariant Metric ◽  
Translation Invariant ◽  
Basic Properties ◽  
The Family ◽  
Cancellation Theorem

AbstractIn the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

Maximal linked systems on families of measurable rectangles

2021 ◽  
pp. 77-104
Author(s):  
Aleksandr G. Chentsov
Keyword(s):  
Probability Theory ◽  
Measure Theory ◽  
Cartesian Product ◽  
Wide Sense ◽  
The Family ◽  
Principal Property ◽  
Measurable Structure ◽  
Stone Type ◽  
By Products ◽  
Algebra Of Sets

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

scholarly journals Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 417 ◽  
Author(s):  
Hu Zhao ◽  
Hong-Ying Zhang
Keyword(s):  
Algebraic Structure ◽  
Rough Sets ◽  
Lattice Structure ◽  
Single Domain ◽  
One Dimensional ◽  
Basic Properties ◽  
Approximation Operators ◽  
Rough Approximation ◽  
The One ◽  
Comprehensive Study

As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved.

scholarly journals Cartesian product and Topology On Fuzzy BI-Algebras

2021 ◽  
Vol 8 ◽  
pp. 29-33
Author(s):  
Gerima Tefera ◽  
Abdi Oli
Keyword(s):  
Cartesian Product ◽  
Fuzzy Topology ◽  
Basic Properties ◽  
And Topology

In this paper,the concepts of homomorphism in fuzzy BI-algebra is intro- duced, and also basic properties of homo- morphisms are investigated. The cartesian product in fuzzy ideals of BI-algebra is investigated with related prop- erties; The concepts of fuzzy topology on BI- algebra elaborated.

scholarly journals On the representation of metric spaces

1979 ◽  
Vol 20 (3) ◽  
pp. 367-375 ◽  
Author(s):  
G.J. Logan
Keyword(s):  
Algebraic Structure ◽  
Topological Structure ◽  
Dual Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
The Family ◽  
Necessary And Sufficient ◽  
The Relationship

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

On EMV-Semirings

2019 ◽  
Vol 69 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. A. Borzooei ◽  
M. Shenavaei ◽  
A. Di Nola ◽  
O. Zahiri
Keyword(s):  
Distributive Lattice ◽  
Algebraic Structure ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Algebraic Extension ◽  
Theoretic Approach ◽  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Generalized Boolean Algebras

Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

scholarly journals A Novel Approach towards Bipolar Soft Sets and Their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Tahir Mahmood
Keyword(s):  
Decision Making ◽  
Algebraic Structures ◽  
Soft Sets ◽  
New Approach ◽  
Basic Properties ◽  
Novel Approach ◽  
Operational Laws

The notion of bipolar soft sets has already been defined, but in this article, the notion of bipolar soft sets has been redefined, called T-bipolar soft sets. It is shown that the new approach is more close to the concept of bipolarity as compared to the previous ones, and further it is discussed that so far in the study of soft sets and their generalizations, the concept introduced in this manuscript has never been discussed earlier. We have also discussed the operational laws of T-bipolar soft sets and their basic properties. In the end, we have deliberated the algebraic structures associated with T-bipolar soft sets and the applications of T-bipolar soft sets in decision-making problems.

scholarly journals The “Generator” of Int-Soft Filters on Residuated Lattices

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 236 ◽  
Author(s):  
Huarong Zhang ◽  
Minxia Luo
Keyword(s):  
Extension Property ◽  
Residuated Lattices ◽  
Simple Method ◽  
Basic Properties

In this paper, we give the “generator” of int-soft filters and propose the notion of t-int-soft filters on residuated lattices. We study the properties of t-int-soft filters and obtain some commonalities (e.g., the extension property, quotient characteristics, and a triple of equivalent characteristics). We also use involution-int-soft filters as an example and show some basic properties of involution-int-soft filters. Finally, we investigate the relations among t-int-soft filters and give a simple method for judging their relations.

scholarly journals Interval-Valued Fuzzy Soft Graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Onur Zihni ◽  
Yıldıray Çelik ◽  
Güven Kara
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Soft Sets ◽  
Strong Product ◽  
Different Types ◽  
Fuzzy Soft Sets ◽  
Interval Valued

Abstract In this paper, we combine concepts of interval-valued fuzzy soft sets and graph theory. Then we introduce notations of interval-valued fuzzy soft graphs and complete interval-valued fuzzy soft graphs. We also present several different types operations including cartesian product, strong product and composition on interval-valued fuzzy soft graphs and investigate some properties of them.

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