scholarly journals Order alteration in a cascade-ordered set

2020 ◽  
pp. 96-115
Author(s):  
Леонид Александрович Поморцев ◽  
Владимир Иванович Цурков
Keyword(s):  
Relational Databases ◽  
Ordered Set ◽  
Functional Dependence ◽  
Partial Orders ◽  
Ordered Sets ◽  
A Chain

В статье исследуются Частично и Каскадно Упорядоченные Множества (ЧУМ и КУМ) в аспекте развития алгоритмирования в Теории Реляционных Баз Данных (ТРБД). КУМ расширяет понятие ЧУМ за счёт включения в него цепочки вложенных друг в друга Частичных Порядков (ЧП). Примером КУМ служит Последовательность Вывода (ПВ) Функциональной Зависимости (ФЗ) из заданной совокупности ФЗ, в которой имеют место порядки следования и вывода одних ФЗ из других. Необходимость перестройки возникает в случаях повторов ФЗ в ПВ, которые преобразуются одной из Теорем настоящей работы в повторное их использование или, иными словами повторы какой-либо ФЗ заменяются исходящим из него гнездованием. Полученные результаты могут иметь самостоятельное значение в алгебре. В настоящей работе для доказательств применяются так называемые логические схемы. The article examines the Partially and Cascade of Ordered Sets (POS and COS) in aspect of the development of algoritms in the Theory of Relational Databases (TRDB). COS expands the notion of POS by including himself a chain of Partial Orders (PO) wich nested into each other. An example of COS is the Sequence of the Derivation (SD) of the Functional Dependence (FD) from a given set of FDs which have two orders of following and of derivation of some FD from others. The need for restructuring arises in cases of repetitions of the FD in SD, which are transformed into their reuse or, in other words, replicas of any FD are replaced by the outgoing from him nesting. The received results can have an independent value in algebra. In the present paper, so-called logic schemes are used for proofs.

scholarly journals Galois and Pataki Connections on Generalized Ordered Sets

2019 ◽  
pp. 283-323
Author(s):  
Árpád Száz
Keyword(s):  
Present Author ◽  
Ordered Set ◽  
Ordered Sets

In this paper, having in mind Galois and Pataki connections, we establish several basic theorems on increasingly seminormal and semiregular functions between gosets. An ordered pair X(\leq )=(X,\leq ) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set). A function f of one goset X to another Y is called increasingly upper g-seminormal, for some function g of Y to X, if f(x)\leq y implies x \leq g(y). While, the function f is called increasingly upper φ-semiregular, for some function φ of X to itself, if f(u)\leq f(v) implies u\leq \varphi (v). The increasingly lower seminormal (semiregular) functions are defined by the reverse implications. Moreover, a function is called increasingly normal (regular) if it is both increasingly upper and lower seminormal (semiregular). The results obtained extend and supplement several former results of O. Ore and the present author on Galois and Pataki connections. Namely, the pairs (f, g) and (f, φ) may be called increasing Galois and Pataki connections if the function f is increasingly g-normal and φ-regular, respectively.

Topologies of Lattice Products

1966 ◽  
Vol 18 ◽  
pp. 1004-1014 ◽  
Author(s):  
Richard A. Alo ◽  
Orrin Frink
Keyword(s):  
Direct Product ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Order Relation ◽  
Direct Products ◽  
Ordered Sets ◽  
Partially Ordered

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely the ideal topology, the interval topology, and the new interval topology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.

Another Note on Sperner's Lemma

1971 ◽  
Vol 14 (2) ◽  
pp. 255-256 ◽  
Author(s):  
David A. Drake
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Height Function ◽  
Maximum Length ◽  
List Type ◽  
Type Number ◽  
Partially Ordered ◽  
A Chain ◽  
Universal Bounds ◽  
Positive Integers

Let Q be a finite partially ordered (by ≤) set with universal bounds O, I. The height function h of Q is defined by the rule: h(x) is the maximum length of a chain from O to x. Let h(I)=n. Suppose that for each k≥0, there exist positive integers a(k) and b(k) such that all elements of height k(i)are covered by a(k) elements of height k+1;(ii)cover b(k) elements of height k—1.Then we call Q a U-poset. Call a subset S of a partially ordered set an antichain if no two elements of S are comparable.

Some properties about the zero-divisor graphs of quasi-ordered sets

2019 ◽  
Vol 19 (04) ◽  
pp. 2050074
Author(s):  
Junye Ma ◽  
Qingguo Li ◽  
Hui Li
Keyword(s):  
Zero Divisor ◽  
Ordered Set ◽  
Line Graph ◽  
Prime Ideals ◽  
Complete Graphs ◽  
Ordered Sets ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Complete Bipartite Graphs ◽  
Minimal Prime

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.

scholarly journals Retracts of partially ordered sets

1979 ◽  
Vol 27 (4) ◽  
pp. 495-506 ◽  
Author(s):  
Dwight Duffus ◽  
Ivan Rival
Keyword(s):  
Fixed Points ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

AbstractLet P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

scholarly journals Well-order as a construction principle for physical theories

2019 ◽  
Vol 79 (11) ◽  
Author(s):  
Peter Schust
Keyword(s):  
Real Number ◽  
Ordered Set ◽  
Mathematical Concept ◽  
Ordered Sets ◽  
Time And Space ◽  
Fundamental Idea ◽  
Construction Principle ◽  
Number Lines ◽  
Representation Of Time

AbstractPhysics has up to now missed to express in mathematical terms the fundamental idea of events of a path in time and space uniquely succeeding one another. An appropriate mathematical concept that reflects this idea is a well-ordered set. In such a set every subset has a least element. Thus every element of a well-ordered set has as its definite successor the least element of the subset of all elements larger than itself. This is apparently contradictory to the densely ordered real number lines which conventionally constitute the coordinate axes in any representation of time and space and in which between any two numbers exists always another number. In this article it is shown how decomposing this disaccord in favour of well-ordered sets causes spacetime to be discontinuous.

scholarly journals A maximal chain approach to topology and order

1988 ◽  
Vol 11 (3) ◽  
pp. 465-472 ◽  
Author(s):  
R. Vainio
Keyword(s):  
Ordered Set ◽  
Maximal Chain ◽  
Ordered Sets ◽  
Complete Lattices ◽  
Interval Topology ◽  
Convergence Structure ◽  
Order Structures ◽  
General Convergence

On ordered sets (posets, lattices) we regard topologies (or, more general convergence structures) which on any maximal chain of the ordered set induce its own interval topology. This construction generalizes several well-known intrinsic structures, and still contains enough to produce interesting results on for instance compactness and connectedness. The “maximal chain compatibility” between topology (convergence structure) and order is preserved by formation of arbitrary products, at least in case the involved order structures are conditionally complete lattices.

scholarly journals Ideal extensions of ordered sets

2004 ◽  
Vol 2004 (53) ◽  
pp. 2847-2861 ◽  
Author(s):  
Niovi Kehayopulu
Keyword(s):  
Ordered Set ◽  
Ordered Sets ◽  
The Ideal

The ideal extensions of semigroups—without order—have been first considered by Clifford (1950). In this paper, we give the main theorem of the ideal extensions for ordered sets. IfP,Qare disjoint ordered sets, we construct (all) the ordered setsVwhich have an idealP′which is isomorphic toP, and the complement ofP′inVis isomorphic toQ. Conversely, we prove that every extension of an ordered setPby an ordered setQcan be so constructed. Illustrative examples of the main theorem in case of finite ordered sets are given.

Automorphism groups of models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1249-1264 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Closed Subgroup ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Peano Arithmetic ◽  
Ordered Sets ◽  
Torsion Free ◽  
Linearly Ordered Set

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

NONCOMMUTATIVE BROWNIAN MOTIONS INDEXED BY PARTIALLY ORDERED SETS

2010 ◽  
Vol 13 (04) ◽  
pp. 629-661 ◽  
Author(s):  
ANNA KULA ◽  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorems ◽  
Ordered Set ◽  
Positive Cone ◽  
Symmetric Matrices ◽  
Ordered Sets ◽  
Brownian Motions ◽  
Partially Ordered ◽  
Independent Increments ◽  
Lorentz Cone ◽  
The Given

We construct noncommutative Brownian motions indexed by partially ordered subsets of Euclidean spaces. The noncommutative independence under consideration is the bm-independence and the time parameter is taken from a positive cone in a vector space ([Formula: see text], the Lorentz cone or the positive definite real symmetric matrices). The construction extends the Muraki's idea of monotonic Brownian motion. We show that our Brownian motions have bm-independent increments for bm-ordered intervals. The appropriate version of the Donsker Invariance Principle is also proved for each positive cone. It requires the bm-Central Limit Theorems related to intervals in the given partially ordered set of indices.

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