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.

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.

Hechler's Theorem for tall analytic P-ideals

2011 ◽  
Vol 76 (2) ◽  
pp. 729-736
Author(s):  
Barnabás Farkas
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Countable Subset ◽  
Generic Extension ◽  
Classical Theorem ◽  
Ground Model ◽  
Partially Ordered

AbstractWe prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal (coded in the ground model) a cofinal subset of is order isomorphic to (Q, ≤).

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.

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.

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.

Real functions on the family of all well-ordered subsets of a partially ordered set

1983 ◽  
Vol 48 (1) ◽  
pp. 91-96 ◽  
Author(s):  
Stevo Todorčević
Keyword(s):  
Set Theory ◽  
Real Function ◽  
Partially Ordered Set ◽  
Initial Segment ◽  
Ordered Set ◽  
Monotone Mapping ◽  
Ordered Sets ◽  
Partially Ordered ◽  
The Family ◽  
Aronszajn Trees

Definition 1 (Kurepa [3, p. 99]). Let E be a partially ordered set. Then σE denotes the set of all bounded well-ordered subsets of E. We consider σE as a partially ordered set with ordering defined as follows: st if and only if s is an initial segment of t.Then σE is a tree, i.e., {s ∈ σ E∣ st} is well-ordered for every t ∈ σE. The trees of the form αE were extensively studied by Kurepa in [3]–[10]. For example, in [4], he used σQ and σR to construct various sorts of Aronszajn trees. (Here Q and R denote the rationals and reals, respectively.) While considering monotone mapping between some kind of ordered sets, he came to the following two questions several times:P.1. Does there exist a strictly increasing rational function on σQ? (See [4, Problème 2], [5, p. 1033], [6, p. 841], [7, Problem 23.3.3].)P.2. Let T be a tree in which every chain is countable and every level has cardinality <2ℵ0. Does there exist a strictly increasing real function on T? (See [6, p. 246] and [7].)It is known today that Problem 2 is independent of the usual axioms of set theory (see [1]). Concerning Problem 1 we have the following.

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{&#xF01C;}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{&#xF01C;}}$. It is shown that ${\cal D}_{\rm{&#xF01C;}} $ contains an antichain of size $2^{\rm{&#xF01C;}} $, and so has size $2^{\rm{&#xF01C;}} $. 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.

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

Summarizing data using partially ordered set theory: An application to fiscal frameworks in 97 countries

2016 ◽  
Vol 32 (3) ◽  
pp. 383-402
Author(s):  
Julia Bachtrögler ◽  
Harald Badinger ◽  
Aurélien Fichet de Clairfontaine ◽  
Wolf Heinrich Reuter
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Fiscal Frameworks
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