On better-quasi-ordering transfinite sequences

1968 ◽  
Vol 64 (2) ◽  
pp. 273-290 ◽  
Author(s):  
C. St. J. A. Nash-Williams
Keyword(s):  
Ordered Set ◽  
Ordinal Number ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Transitive Relation ◽  
Ordinal Numbers ◽  
One To One ◽  
Quasi Ordering

AbstractLet Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

On well-quasi-ordering transfinite sequences

1965 ◽  
Vol 61 (1) ◽  
pp. 33-39 ◽  
Author(s):  
C. St. J. A. Nash-Williams
Keyword(s):  
Finite Subset ◽  
Ordered Set ◽  
Infinite Sequence ◽  
Infinite Subset ◽  
Classical Theorem ◽  
Transitive Relation ◽  
Finite Sequences ◽  
One To One ◽  
Positive Integers ◽  
Quasi Ordering

Abstract. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i < j and qi ≤ qi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write f ≤ g. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, t∈A(P), write s ≺ t if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that s ≺ t. Then there exists an infinite subset I of P such that T ∩ A(I)is contained in a single Tj.

On well-quasi-ordering infinite trees

1965 ◽  
Vol 61 (3) ◽  
pp. 697-720 ◽  
Author(s):  
C. St. J. A. Nash-Williams
Keyword(s):  
Ordered Set ◽  
Infinite Sequence ◽  
Transitive Relation ◽  
Infinite Trees ◽  
Finite Sequences ◽  
Positive Integers ◽  
Quasi Ordering

AbstractLet A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and s is x1, …, xm and t is x2, x3, …, xn. Call a subset S of A a P-block if, for every infinite ascending sequence x1, x2, … of positive integers, there exists an m such that x1, …, xm belongs to S. A quasi-ordered set Q (i.e. a set on which a reflexive and transitive relation ≤ is defined) is better-quasi-ordered if, for every P-block S and every function f:S → Q, there exist s, t ∈ S such that s ⊲ t and f(s) ≤ f(t). It is proved that any set of (finite or infinite) trees is better-quasi-ordered if T1 ≤ T2 means that the tree T1 is homeomorphic to a subtree of the tree T2. This establishes a conjecture of J. B.Kruskal that, if T1, T2, … is an infinite sequence of trees, then there exist i, j such that i < j and Ti ≤ Tj.

Formal development of ordinal number theory

1955 ◽  
Vol 20 (2) ◽  
pp. 95-104
Author(s):  
Steven Orey
Keyword(s):  
Number Theory ◽  
Ordinal Number ◽  
Ordered Sets ◽  
Stringent Condition ◽  
Easy Consequence ◽  
Basic Properties ◽  
Ordinal Numbers ◽  
Order Types ◽  
Proper Subclass ◽  
Definition Of

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

Weakly Polymatroidal Ideals

2006 ◽  
Vol 13 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Masako Kokubo ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Partially Ordered Sets ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Fundamental Fact ◽  
Polymatroidal Ideals ◽  
Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

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.

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.

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