scholarly journals WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

2018 ◽  
Vol 83 (1) ◽  
pp. 392-394
Author(s):  
LINDA BROWN WESTRICK
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Turing Degrees ◽  
Image Position ◽  
Scott Set

AbstractLet ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

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.

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 SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

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.

scholarly journals THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

scholarly journals Scott's problem for Proper Scott sets

2008 ◽  
Vol 73 (3) ◽  
pp. 845-860 ◽  
Author(s):  
Victoria Gitman
Keyword(s):  
Boolean Algebra ◽  
Partial Order ◽  
Standard System ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Scott Set ◽  
Forcing Axiom

AbstractSome 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set is proper if the quotient Boolean algebra /Fin is a proper partial order and A-proper if is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.

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.

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