scholarly journals Injective and projective semimodules over involutive semirings

2021 ◽  
pp. 2250182
Author(s):  
Peter Jipsen ◽  
Sara Vannucci
Keyword(s):  
Crucial Role ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Term Equivalence ◽  
Involutive Residuated Lattice ◽  
Necessary And Sufficient ◽  
Mv Algebras

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [Formula: see text] to be a subalgebra of an involutive residuated lattice, where [Formula: see text] is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

The existence of states on EQ-algebras

2020 ◽  
Vol 70 (3) ◽  
pp. 527-546
Author(s):  
Xiao Long Xin ◽  
Ying Cang Ma ◽  
Yu Long Fu
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Open Problems ◽  
Fantastic Filter ◽  
Necessary And Sufficient ◽  
Bosbach State

AbstractInspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.

Fuzzy Regular Languages Based on Residuated Lattice

2020 ◽  
Vol 16 (02) ◽  
pp. 363-376
Author(s):  
Anupam K. Singh ◽  
S. P. Tiwari
Keyword(s):  
Residuated Lattice ◽  
Finite Automata ◽  
Regular Languages ◽  
Sufficient Condition ◽  
Membership Functions ◽  
Necessary And Sufficient Condition ◽  
Complete Residuated Lattice ◽  
Pumping Lemma ◽  
Fuzzy Finite Automata ◽  
Necessary And Sufficient

The purpose of this work is to introduce the concept of fuzzy regular languages (FRL) accepted by fuzzy finite automata, and try to introduce the categorical look of fuzzy languages where the codomain of membership functions are taken as a complete residuated lattice, instead of [Formula: see text]. Also, we have introduced pumping lemma for FRL, which is used to establish a necessary and sufficient condition for a given fuzzy languages to be non-constant.

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.

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