A study on distributive and modular lattice ordered fuzzy soft group and its duality

Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft group is the generalization of soft group. Abdulkadir Aygunoglu and Halis Aygun introduced the notion of fuzzy soft groups in 2009[1]. In this paper, the concept of lattice ordered fuzzy soft groups and its duality has been introduced. Then distributive and modular lattice ordered fuzzy soft groups are analysed. The objective of this paper is to study the lattice theory over the collection of fuzzy soft group in a parametric manner. Some pertinent properties have been analysed and hence established duality principle.

REVERSE ORDER LAW FOR GROUP INVERSE OF PRODUCT OF MATRICES ON SKEW FIELDS

Let Kn×n be the set of all n×n matrices and Krn×n the set {A∈Kn×n|rankA=r} on askew field K. Zhuang [1] denotes by A# the group inverse of A∈Kn×n which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.

The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory

Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg1 ≤ hg2 whenever g1 ≤ g2. G is orderable provided it admits a total order ≤ satisfying both: hg1 ≤ hg2 whenever g1 ≤ g2 and g1h ≤ g2h whenever g1 ≤ g2. A classical result shows that free groups are orderable. In this paper, we prove that left-orderable groups and orderable groups are quasivarieties of groups both with undecidable theory. For orderable groups, we find an explicit set of universal axioms.

Maximal Above and Below Pairs of Abelian Fully Ordered Groups

In this paper, maximal above and below pairs of fully ordered groups (o groups) are studied. It is shown that maximal above and below pairs of an divisible abelian o group have some specific properties, and by using these properties another embedding method of an abelian o group to real valued functions is discussed.

Endpoints of Multi-Valued Weak Contractions on the Metric Space of Partially Ordered Groups

The present work considers the endpoint in the abstract metric space. It firstly introduces the metric space of partially ordered groups and the metric space of partially ordered modules, respectively;and defines the convergence of sequences and the multi-valued weak contractions, etc., on the introduced space. And then, with the methods of functional analysis and abstract algebra, it successively establishes an endpoint theorem for the metric space of partially ordered groups and an endpoint theorem for the metric space of partially ordered modules. The contributions of this article extend the theory of cone metric space constructed by Huang and Zhang (2007) and some recent results on the fixed point and endpoint theory, such as the endpoint theorem given by Amini-Harandi (2010).

Evaluate All the Order of Every Element in the Higher Order of Group for Addition and Multiplication Composition

This paper aims at treating a study on the order of every element for addition and multiplication composition in the higher order of groups for different algebraic structures as groups;order of a group and order of element of a group in real numbers. Here we discuss the higher order of groups in different types of order which will give us practical knowledge to see the applications of the addition and multiplication composition. If G is a finite group, n is a positive integer and a ⋴G, then the order of the products na. When G is a finite group, every element must have finite order. However, the converse is false: there are infinite groups where each element has finite order. For example, in the group of all roots of unity in C× each element has finite order. Finally, we find out the order of every element of a group in different types of higher order of group.

K_0-group Transmissibilities of the Tracial Limit of C*-algebras

We introduce the tracial limit A = （t4）lim n→∞（An,pn） and show that if K0（An） has ordered relation, K0（A） has ordered relation naturally. In the case that A is simple and K0（An） is weakly unperforated for every n, K0（A） is weakly unperforated too. Furthermore, the Riesz interpolation property of K0（An） can be transmitted to K0（A）.

K-THEORETICALLY COMPLETE VON NEUMANN REGULAR RINGS

The authors introduce the concept of Ko-complete ring and study the properties of Ko-complete unit-regular rings.

IBN Rings and Orderings on Grothendieck Groups

Let R be a ring with an identity element.R∈IBN means that Rm■Rn implies m=n,R ∈IBN1 means that Rm ■Rn⊕K implies m≥n,and R ∈IBN2 means that Rm■Rm⊕K implies K=0.In this paper we give some characteristic properties of IBN1 and IBN2,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM1 and all finitely generated projective left R-modules are stably free,then the Grothendieck group Ko（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group Ko（R）of a ring R is a partial ordering,then R ∈IBM1 or Ko（R）=0.

Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.

Construction of mesoporous ceria-supported gold catalysts with rich oxygen vacancies for efficient CO oxidation

Bearing unique redox nature and high oxygen storage capacity,ceria(CeO_(2))has always been a promising CO oxidation catalyst support for gold(Au)catalysts and the like.Herein,a series of Au-CeO_(2)-P(P stands for pH value)samples was prepared by a co-precipitation method with the assistance of an alkaline environment and amino groups functionalized ordered mesoporous polymer(OMP-NH_(2)).Afterward,all samples described above were characterized that the Au-CeO_(2)-P catalysts are made of Au-Ce-O solid solution and Au nanoparticles(NPs)supported on CeO_(2).It turns out that OMP-NH_(2) is not just a simple sacrificial template for mesoporous structure,but also plays an important role as an amino source,explaining the presence of rich oxygen vacancies.Due to the concentration of oxygen vacancies in Au-Ce-O solid solution is the key factor for the oxygen mobility of CO oxidation,the catalytic results also demonstrate that the catalytic activity of Au-CeO_(2)-P catalysts is related to the concentration of their oxygen vacancies.Moreover,Au-CeO_(2)-9.6 with a highest concentration of oxygen vacancies(as high as 13.98%)in Au-CeO_(2)-P catalysts exhibits the best catalytic activity(complete conversion at 10℃).

Blowing-ups and Valuations on Surfaces

In this paper, we show that all the nontrivial valuations on surfaces can be given by the infinite sequences of blowing-ups, and give the process of blowing-ups.