THE TUKEY ORDER ON COMPACT SUBSETS OF SEPARABLE METRIC SPACES

2016 ◽  
Vol 81 (1) ◽  
pp. 181-200 ◽  
Author(s):  
PAUL GARTSIDE ◽  
ANA MAMATELASHVILI
Keyword(s):  
Partially Ordered Set ◽  
Metric Spaces ◽  
Ordered Set ◽  
Equivalence Classes ◽  
Order Structure ◽  
Partially Ordered ◽  
Image Position ◽  
Separable Metrizable Space ◽  
Directed Sets ◽  
Compact Subsets

AbstractOne partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{}}$. It is shown that ${\cal D}_{\rm{}} $ contains an antichain of size $2^{\rm{}} $, and so has size $2^{\rm{}} $. The elements of the antichain are of the form ${\cal K}\left( M \right)$, the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.

Maximal or Greatest Homomorphic Images of Given Type

1968 ◽  
Vol 20 ◽  
pp. 264-271 ◽  
Author(s):  
Takayuki Tamura
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Image Position

Let Q be a quasi-ordered set with respect to ⩽ ; that is, the order ⩽ is reflexive and transitive. An element a of Q is called maximal (minimal) ifa is called greatest (smallest) ifObviously a greatest (smallest) element is maximal (minimal). A greatest (smallest) element in a partially ordered set is unique, but it is not necessarily unique in a quasi-ordered set.

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

Completeness in Semi-Lattices

1957 ◽  
Vol 9 ◽  
pp. 578-582 ◽  
Author(s):  
L. E. Ward
Keyword(s):  
Binary Relation ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Image Position ◽  
Transitive Binary Relation

Let (X, ≤) be a partially ordered set, that is, X is a set and ≤ is a reflexive, anti-symmetric, transitive, binary relation on X.We write,for each x ∈ X. If, moreover,exists for each x and y in X, then (X, ≤) is said to be a semi-lattice.

scholarly journals Pointwise compact and stable sets of measurable functions

1993 ◽  
Vol 58 (2) ◽  
pp. 435-455 ◽  
Author(s):  
S. Shelah ◽  
D. H. Fremlin
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Measure Theory ◽  
Compact Set ◽  
Compact Sets ◽  
Stable Sets ◽  
Partially Ordered ◽  
Measurable Functions ◽  
Image Position

In a series of papers culminating in [9], M. Talagrand, the second author, and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is ‘stable’ in Talagrand's sense, the point being that stable sets are known to have a variety of properties not shared by all pointwise compact sets. In the present paper we present a model of set theory in which there is a countable relatively pointwise compact set of Lebesgue measurable functions which is not stable and discuss the significance of this model in relation to the original questions. A feature of our model which may be of independent interest is the following: in it, there is a closed negligible set Q ⊆ [0, 1]2 such that whenever D ⊆ [0,1] has outer measure 1, thenhas inner measure 1 (see 2G below).We embark immediately on the central ideas of this paper, setting out a construction of a partially ordered set which forces a fairly technical proposition in measure theory (IS below); the relevance of this proposition to pointwise compact sets will be discussed in §2.

The consistency of the axiom of universality for the ordering of cardinalities

1985 ◽  
Vol 50 (2) ◽  
pp. 502-509
Author(s):  
Marco Forti ◽  
Furio Honsell
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Regular Cardinal ◽  
Ordered Set ◽  
Partial Ordering ◽  
Generic Extension ◽  
Ground Model ◽  
Partially Ordered ◽  
Transfer Method ◽  
Image Position

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.

Abian's poset and the ordered monoid of annihilator classes in a reduced commutative ring

2014 ◽  
Vol 13 (08) ◽  
pp. 1450070 ◽  
Author(s):  
David F. Anderson ◽  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Equivalence Relation ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Equivalence Classes ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Ordered Monoid

Let R be a reduced commutative ring with 1 ≠ 0. Then R is a partially ordered set under the Abian order defined by x ≤ y if and only if xy = x2. Let RE be the set of equivalence classes for the equivalence relation on R given by x ~ y if and only if ann R(x) = ann R(y). Then RE is a commutative Boolean monoid with multiplication [x][y] = [xy] and is thus partially ordered by [x] ≤ [y] if and only if [xy] = [x]. In this paper, we study R and RE as both monoids and partially ordered sets. We are particularly interested in when RE can be embedded in R as either a monoid or a partially ordered set.

The space of formal balls and models of quasi-metric spaces

2009 ◽  
Vol 19 (2) ◽  
pp. 337-355 ◽  
Author(s):  
M. ALI-AKBARI ◽  
B. HONARI ◽  
M. POURMAHDIAN ◽  
M. M. REZAII
Keyword(s):  
Metric Space ◽  
Partially Ordered Set ◽  
Metric Spaces ◽  
Ordered Set ◽  
Space Problem ◽  
Partially Ordered ◽  
Sorgenfrey Line ◽  
Maximal Point ◽  
Ideal Completion ◽  
The Ideal

In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, ⊑, ≺). We prove that if the conjugate space (X, d−1) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, ⊑, ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

scholarly journals A Theorem on Partially Ordered Sets and its Application

1969 ◽  
Vol 9 (3-4) ◽  
pp. 361-362
Author(s):  
Vladimir Devidé
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Image Position

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

scholarly journals Addendum to a paper of M. Saks

1984 ◽  
Vol 25 (1) ◽  
pp. 31-33 ◽  
Author(s):  
Hazel Perfect
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Maximum Cardinality ◽  
Partially Ordered ◽  
Image Position

Throughout, (X, ≤ ) denotes a partially ordered set (p. o. set), where X is assumed to be finite. A subset Y of X is called a k-union if Y contains no chain of length K + 1. In particular, therefore, a 1-union is just an antichain; and it is readily seen that Y is a k-union if and only if it is a union of K antichains. (Dually, a subset Z of X is a k-counion if Z contains no antichain of length k + 1.) We denote by dk (X) the maximum cardinality of a k-union in X, with a similar notation for other p. o. sets. Now let be any partition of X into chains, and write.

ON RAMSEY’S THEOREM AND THE EXISTENCE OF INFINITE CHAINS OR INFINITE ANTI-CHAINS IN INFINITE POSETS

2016 ◽  
Vol 81 (1) ◽  
pp. 384-394 ◽  
Author(s):  
ELEFTHERIOS TACHTSIS
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Infinite Chain ◽  
Ordered Sets ◽  
Ramsey's Theorem ◽  
Partially Ordered ◽  
Ramsey’S Theorem ◽  
Image Position ◽  
The Axiom Of Choice

AbstractRamsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle.In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel–Mostowski permutation model ${\cal N}$ for ZFA and prove that the above principle for infinite partially ordered sets is true in ${\cal N}$, whereas Ramsey’s Theorem is false in ${\cal N}$. Then, based on the existence of ${\cal N}$ and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem.In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true in Mostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.

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