scholarly journals CLOSURE OPERATORS, FRAMES AND NEATEST REPRESENTATIONS

2017 ◽  
Vol 96 (3) ◽  
pp. 361-373
Author(s):  
ROB EGROT
Keyword(s):  
Closure Operator ◽  
Sufficient Condition ◽  
Recursive Construction ◽  
Necessary And Sufficient Condition ◽  
Closure Operators ◽  
Closed Sets ◽  
Necessary And Sufficient

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$, we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$-closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$-closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$, the lattice of ${\mathcal{U}}$-ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$-distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$. From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(\unicode[STIX]{x1D714},3)$-representation.

scholarly journals On the Equationally Artinian Groups

2020 ◽  
pp. 583-595
Author(s):  
Mohammad Shahryari ◽  
Javad Tayyebi
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Finite Extension ◽  
Finite Union ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Sets ◽  
Necessary And Sufficient

In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups

scholarly journals On topological soft sets

2020 ◽  
Vol 16 (2) ◽  
pp. 5-22
Author(s):  
M. Burç Kandemir ◽  
B. Tanay
Keyword(s):  
Topological Space ◽  
Structural Properties ◽  
Soft Set ◽  
Point Of View ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Soft Sets ◽  
Closed Sets ◽  
Necessary And Sufficient

AbstractIn this paper, we have established topological soft sets over generalized topological spaces and topological spaces, and studied its structural properties. We have derived a topological soft set in any given topological space, and from this point of view, we have given necessary and sufficient condition for homeomorphic Alexandroff spaces using topological soft set technique. At last, we have derived a topological soft set using closed sets in any topological space and we have given necessary and sufficient condition for arbitrary homeomorphic topological spaces using them.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

Sign in / Sign up

Export Citation Format

Share Document