scholarly journals Complex Intuitionistic Fuzzy Soft Lattice Ordered Group and Its Weighted Distance Measures

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 705 ◽  
Author(s):  
S. Rajareega ◽  
J. Vimala ◽  
D. Preethi
Keyword(s):  
Set Theory ◽  
Euclidean Distance ◽  
Distance Measures ◽  
Fuzzy Soft Set ◽  
Ordered Group ◽  
Weighted Distance ◽  
Intuitionistic Fuzzy ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Algebraic Properties

In recent years, the complex fuzzy set theory has intensified the attention of many researchers. This paper focuses on developing the algebraic structures pertaining to lattice ordered groups and lattice ordered subgroups for complex intuitionistic fuzzy soft set theory. Furthermore, some of their properties and operations are discussed. In addition, the weighted distance measures between two complex intuitionistic fuzzy soft lattice ordered groups such as weighted hamming, weighted normalized hamming, weighted euclidean and weighted normalized euclidean distance measures were introduced and also some of the algebraic properties of the weighted distance measures are verified. Moreover, the application of complex intuitionistic fuzzy soft lattice ordered groups by using the weighted distance measures is analysed.

Operations on complex intuitionistic fuzzy soft lattice ordered group and CIFS-COPRAS method for equipment selection process

2021 ◽  
pp. 1-10
Author(s):  
S. Rajareega ◽  
J. Vimala
Keyword(s):  
Selection Process ◽  
Lattice Ordered Group ◽  
Fuzzy Soft Set ◽  
Ordered Group ◽  
Equipment Selection ◽  
Intuitionistic Fuzzy ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Fuzzy Lattice ◽  
Copras Method

This paper introduces some new operations on complex intuitionistic fuzzy lattice ordered groups such as sum, product, bounded product, bounded difference and disjoint sum, and verifying its pertinent properties. The research exhibits the CIFS-COPRAS algorithm in a complex intuitionistic fuzzy soft set environment. This method was furthermore applied for the equipment selection process.

The Hahn representation theorem for ℓ-groups in ZFA

2000 ◽  
Vol 65 (2) ◽  
pp. 519-524
Author(s):  
D. Gluschankof
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Group Structure ◽  
Ordered Group ◽  
Representation Theorems ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Totally Ordered Group ◽  
Basic Concepts ◽  
The Axiom Of Choice

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

Some Intuitionist Fuzzy Weighted Geometric Distance Measures and Their Application to Group Decision Making

2014 ◽  
Vol 22 (05) ◽  
pp. 699-715 ◽  
Author(s):  
Bo Peng ◽  
Chunming Ye ◽  
Shouzhen Zeng
Keyword(s):  
Decision Making ◽  
Group Decision Making ◽  
Group Decision ◽  
Hamming Distance ◽  
Distance Measures ◽  
Weighted Distance ◽  
Geometric Distance ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Environment ◽  
Consensus Reaching Process

The ordered weighted distance (OWD) measure developed by Xu and Chen having been proved suitable to deal with the situation where the input arguments are represented in exact numerical values. In this paper, we develop some new geometric distance measures with intuitionistic fuzzy information, which are the generalization of some widely used distance measures, including the intuitionistic fuzzy weighted geometric distance (IFWGD) measure, the intuitionistic fuzzy ordered weighted geometric distance (IFOWGD) measure, the intuitionistic fuzzy ordered weighted geometric Hamming distance (IFOWGHD) measure, the intuitionistic fuzzy ordered weighted geometric Euclidean distance (IFOWGED) measure, the intuitionistic fuzzy hybrid weighted geometric distance (IFHWGD) measure. These developed weighted geometric distance measures are very suitable to deal with the situation where the input arguments are represented in intuitionistic fuzzy values. And then, we present a consensus reaching process based on the developed distance measures with intuitionistic fuzzy preference information for group decision making. Finally, we apply the developed approach with a numerical example to group decision making under intuitionistic fuzzy environment.

Archimedean Closures in Lattice-Ordered Groups

1969 ◽  
Vol 21 ◽  
pp. 1004-1012 ◽  
Author(s):  
Richard D. Byrd
Keyword(s):  
Partial Answer ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Regular Subgroup

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

scholarly journals Tight Riesz groups

1972 ◽  
Vol 13 (2) ◽  
pp. 224-240 ◽  
Author(s):  
R. J. Loy ◽  
J. B. Miller
Keyword(s):  
General Theory ◽  
Open Interval ◽  
Discrete Topology ◽  
Topological Groups ◽  
Ordered Group ◽  
Partially Ordered ◽  
Interval Topology ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Positive Elements

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

On a relative uniform completion of an archimedean lattice ordered group

2009 ◽  
Vol 59 (2) ◽  
Author(s):  
Štefan Černák ◽  
Judita Lihová
Keyword(s):  
Uniform Convergence ◽  
Subdirect Product ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice ◽  
Finite Direct Product ◽  
Relatively Uniform Convergence

AbstractThe notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups.Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) $$ G_{\omega _1 } $$ of G is dealt with. It is known that $$ G_{\omega _1 } $$ exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in $$ G_{\omega _1 } $$. Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.

Relatively uniform convergences in archimedean lattice ordered groups

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Ján Jakubík ◽  
Štefan Černák
Keyword(s):  
Uniform Convergence ◽  
Vector Lattice ◽  
Radical Class ◽  
Ordered Group ◽  
Dedekind Completion ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Divisible Hull ◽  
Archimedean Lattice ◽  
Relatively Uniform Convergence

AbstractFor an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

scholarly journals C*-algebras generated by isometries and true representations

2019 ◽  
Vol 38 (5) ◽  
pp. 215-232
Author(s):  
Mamoon Ahmed
Keyword(s):  
The Other ◽  
Lattice Ordered Group ◽  
Covariant Representation ◽  
Ordered Group ◽  
Two Stage ◽  
C Algebras ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

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