Separation axioms for topological ordered spaces

1968 ◽  
Vol 64 (4) ◽  
pp. 965-973 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Separation Axioms ◽  
Ordered Space

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.

Partial Orders on the 2-Cell

1975 ◽  
Vol 18 (3) ◽  
pp. 411-416
Author(s):  
E. D. Tymchatyn
Keyword(s):  
Partial Order ◽  
Metric Space ◽  
Closed Subset ◽  
Cartesian Product ◽  
Partial Orders ◽  
Partial Ordering ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Ordered Space

A partially ordered space is an ordered pair (X, ≤) where X is a compact metric space and ≤ is a partial ordering on X such that ≤ is a closed subset of the Cartesian product X×X. ≤ is said to be a closed partial order on X.

A quotient ordered space

1968 ◽  
Vol 64 (2) ◽  
pp. 317-322 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Partial Orders ◽  
Projection Mapping ◽  
Quotient Spaces ◽  
The Family ◽  
Ordered Space ◽  
Image Position

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the propertya partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

Ordering Uniform Completions of Partially Ordered Sets

1974 ◽  
Vol 26 (3) ◽  
pp. 644-664 ◽  
Author(s):  
R. H. Redfield
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Partial Orders ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Ordered Space ◽  
Uniformly Continuous

Let (P, ) be a (nearly) uniform ordered space. Let (P, ) be the uniform completion of (P, ) at . Several partial orders for P are introduced and discussed. One of these orders provides an adjoint to the functor which embeds the category of uniformly complete uniform ordered spaces in the category of uniform ordered spaces, both categories with uniformly continuous order-preserving functions. When P is a join-semilattice with uniformly continuous join, two of these orders coalesce to the unique partial order with respect to which P is a join-semilattice, P is a join-subsemilattice of P, and the join on P is continuous.

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.

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 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.

On a generalization of Jensen's □κ, and strategic closure of partial orders

1983 ◽  
Vol 48 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Recent Work ◽  
Partial Orders ◽  
Closure Condition ◽  
Forcing Axioms ◽  
Combinatorial Principles ◽  
The Relationship ◽  
Mahlo Cardinals ◽  
Forcing Axiom

It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □κ. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □κ.In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈P, ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈pβ∣β < α〉 Good chooses all terms pβ for limit β, and Bad chooses all the others. Bad wins if for some limit β<α, Good is unable to move at stage β because 〈pγ∣γ < β〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good.

scholarly journals Core partial order in rings with involution

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5695-5701 ◽  
Author(s):  
Xiaoxiang Zhang ◽  
Sanzhang Xu ◽  
Jianlong Chen
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Reverse Order ◽  
Unital Ring ◽  
Reverse Order Law ◽  
Rings With Involution ◽  
Invertible Elements

Let R be a unital ring with involution. Several characterizations and properties of core partial order in R are given. In particular, we investigate the reverse order law (ab)# = b#a# for two core invertible elements a, b ? R. Some relationships between core partial order and other partial orders are obtained.

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