scholarly journals Extendible spaces

2002 ◽  
Vol 3 (2) ◽  
pp. 169 ◽  
Author(s):  
M.P. Schellekens
Keyword(s):  
Partial Order ◽  
Metric Spaces ◽  
Extension Property ◽  
Partial Orders ◽  
Domain Theory ◽  
Extremal Element

<p>The domain theoretic notion of lifting allows one to extend a partial order in a trivial way by a minimum. In the context of Quantitative Domain Theory partial orders are represented as quasi-metric spaces. For such spaces, the notion of the extension by an extremal element turns out to be non trivial.</p><p>To some extent motivated  by these considerations, we characterize the directed quasi-metric spaces extendible by an extremum. The  class is shown to include the S-completable directef quasi-metric spaces. As an application of this result, we show that for the case of the invariant quasi-metric (semi)lattices, weightedness can be characterized by order convexity with the extension property.</p>

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

scholarly journals Preface

2015 ◽  
Vol 27 (4) ◽  
pp. 459-459
Author(s):  
ACHIM JUNG ◽  
GUO-QIANG ZHANG
Keyword(s):  
Lattice Theory ◽  
Process Algebra ◽  
Metric Spaces ◽  
Domain Theory ◽  
Conference Series ◽  
The Third ◽  
Language Semantics ◽  
And Mathematics ◽  
October 17 ◽  
Partial Logics

The International Symposium on Domain Theory (ISDT) is a conference series intended to be a forum for researchers in domain theory and its applications. Topics include topological and logical aspects of domains; categories of domains and powerdomains; continuous posets and their representations; partial orders, lattice theory and metric spaces; types, process algebra and concurrency; non-classical and partial logics; programming language semantics; applications in computer science and mathematics. This conference series was founded by Yingming Liu, Yixiang Chen, Klaus Keimel, and Guo-Qiang Zhang. All ISDT events have taken place in China. The first ISDT was held in Shanghai, October 17–24, 1999; the second ISDT was held in Chengdu, October 22–26, 2001; the third ISDT occurred in Xi'an, China, May 10–14, 2004; the fourth ISDT was held in Changsha, June 2–6, 2006; and the fifth ISDT took place in Shanghai, September 11–14, 2009.

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

scholarly journals A Study of Completely Inverse Paramedial AG-Groupoids

2021 ◽  
pp. 101-115
Author(s):  
Muhammad Rashad ◽  
Imtiaz Ahmad ◽  
Faruk Karaaslan
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Left Identity ◽  
Normal Congruence ◽  
Paramedial Law ◽  
Congruence Pair

A magma S that meets the identity, xy·z = zy·x, ∀x, y, z ∈ S is called an AG-groupoid. An AG-groupoid S gratifying the paramedial law: uv · wx = xv · wu, ∀ u, v, w, x ∈ S is called a paramedial AGgroupoid. Every AG-grouoid with a left identity is paramedial. We extend the concept of inverse AG-groupoid [4, 7] to paramedial AG-groupoid and investigate various of its properties. We prove that inverses of elements in an inverse paramedial AG-groupoid are unique. Further, we initiate and investigate the notions of congruences, partial order and compatible partial orders for inverse paramedial AG-groupoid and strengthen this idea further to a completely inverse paramedial AG-groupoid. Furthermore, we introduce and characterize some congruences on completely inverse paramedial AG-groupoids and introduce and characterize the concept of separative and completely separative ordered, normal sub-groupoid, pseudo normal congruence pair, and normal congruence pair for the class of completely inverse paramedial AG-groupoids. We also provide a variety of examples and counterexamples for justification of the produced results.

scholarly journals Some fixed point theorems via partial order relations without the monotone property

2015 ◽  
Vol 31 (3) ◽  
pp. 389-394
Author(s):  
WARUT SAKSIRIKUN ◽  
◽  
NARIN PETROT ◽  
Keyword(s):  
Fixed Point ◽  
Partial Order ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Coupled Fixed Point ◽  
Order Relation ◽  
Monotone Mappings ◽  
Complete Metric Spaces ◽  
Nonlinear Anal ◽  
Monotonic Properties

The main aim of this paper is to consider some fixed point theorems via a partial order relation in complete metric spaces, when the considered mapping may not satisfy the monotonic properties. Furthermore, we also obtain some couple fixed point theorems, which can be viewed as an extension of a result that was presented in [V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 7347–7355].

scholarly journals Some Globally Stable Fixed Points in b-Metric Spaces

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 555 ◽  
Author(s):  
Umar Batsari ◽  
Poom Kumam ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Fixed Points ◽  
Partial Order ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Continuous Mapping ◽  
Regularity Property ◽  
Contractive Mappings ◽  
Metric Function ◽  
Globally Stable

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the b-metric function has the regularity property. Our results improve, and generalize some current results in the literature.

scholarly journals Sobolev Extensions of Lipschitz Mappings into Metric Spaces

2017 ◽  
Vol 2019 (8) ◽  
pp. 2241-2265
Author(s):  
Scott Zimmerman
Keyword(s):  
Heisenberg Group ◽  
Bounded Domain ◽  
Compact Subset ◽  
Metric Space ◽  
Metric Spaces ◽  
Extension Property ◽  
Lipschitz Mapping ◽  
Lipschitz Extension ◽  
Lipschitz Mappings ◽  
Lipschitz Map

Abstract Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C&gt;0$, any $L$-Lipschitz map from a subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ can be extended to a $CL$-Lipschitz mapping on $\mathbb{R}^m$. In this article, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ may be extended to a Sobolev mapping on any bounded domain containing the set. More generally, we prove this result in the case of mappings into any Lipschitz $(n-1)$-connected metric space.

scholarly journals On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

1998 ◽  
Vol 8 (5) ◽  
pp. 481-540 ◽  
Author(s):  
DANIELE TURI ◽  
JAN RUTTEN
Keyword(s):  
Computer Science ◽  
Mathematical Theory ◽  
Metric Spaces ◽  
Operational Semantics ◽  
Denotational Semantics ◽  
Partial Orders ◽  
Present Survey ◽  
Induction Principle ◽  
Structural Operational Semantics ◽  
Modern Mathematics

This paper, a revised version of Rutten and Turi (1993), is part of a programme aiming at formulating a mathematical theory of structural operational semantics to complement the established theory of domains and denotational semantics to form a coherent whole (Turi 1996; Turi and Plotkin 1997). The programme is based on a suitable interplay between the induction principle, which pervades modern mathematics, and a dual, non-standard ‘coinduction principle’, which underlies many of the recursive phenomena occurring in computer science.The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle. The properties of coalgebraic coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders and metric spaces.

scholarly journals Partial Order Rank Features in Colour Space

2020 ◽  
Vol 10 (2) ◽  
pp. 499 ◽  
Author(s):  
Fabrizio Smeraldi ◽  
Francesco Bianconi ◽  
Antonio Fernández ◽  
Elena González
Keyword(s):  
Image Classification ◽  
Partial Order ◽  
Multivariate Data ◽  
Order Relation ◽  
Mathematical Structure ◽  
Partial Orders ◽  
Natural Order ◽  
Colour Space ◽  
Grey Scale

Partial orders are the natural mathematical structure for comparing multivariate data that, like colours, lack a natural order. We introduce a novel, general approach to defining rank features in colour spaces based on partial orders, and show that it is possible to generalise existing rank based descriptors by replacing the order relation over intensity values by suitable partial orders in colour space. In particular, we extend a classical descriptor (the Texture Spectrum) to work with partial orders. The effectiveness of the generalised descriptor is demonstrated through a set of image classification experiments on 10 datasets of colour texture images. The results show that the partial-order version in colour space outperforms the grey-scale classic descriptor while maintaining the same number of features.

Betweenness relations and cycle-free partial orders

1996 ◽  
Vol 119 (4) ◽  
pp. 631-643 ◽  
Author(s):  
J. K. Truss
Keyword(s):  
Partial Order ◽  
Linear Order ◽  
Partial Orders ◽  
Partial Ordering ◽  
Macneille Completion ◽  
Alternative Definition ◽  
Betweenness Relations

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

Sign in / Sign up

Export Citation Format

Share Document