Cofinal Subset Research Articles

A few projective classes of (non-Hausdorff) topological spaces

A class of topological spaces is projective (resp., ω-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, (ω-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even ω-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

APPLICATIONS OF PCF THEORY TO THE STUDY OF IDEALS ON

AbstractLet$\kappa $be a regular uncountable cardinal, anda cardinal greater than or equal to$\kappa $. Revisiting a celebrated result of Shelah, we show that ifis close to$\kappa $and(= the least size of a cofinal subset of) is greater than, thencan be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that ifand, then no$\kappa $-complete ideal onis weakly-saturated.

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets -- CS. $\circ$ Every partially ordered set has a cofinal well-founded subset -- CWF. $\circ$ Dilworth's decomposition theorem for infinite partially ordered sets of finite width -- DT. We also study a graph homomorphism problem and a problem due to A. Hajnal without AC. Further, we study a few statements restricted to linearly-ordered structures without AC.

When Pκ(λ) (vaguely) resembles κ

Let μ<κ<λ be three infinite cardinals, the first two being regular. Assuming the existence of a cofinal subset of (Pκ(λ),⊆) of size λ, we define an ideal on Pκ(λ) which we argue to be the analog for Pκ(λ) of NSκ|Eμκ (the restriction of the nonstationary ideal on κ to the set of limit ordinals less than κ of cofinality μ). We show that this ideal on Pκ(λ) is the ideal dual to the well-known μ-club filter, thus answering a question raised by Krueger and Foreman.

Continuous and other finitely generated canonical cofinal maps on ultrafilters

This paper concentrates on the existence of canonical cofinal maps of three types: continuous, generated by finitary monotone end-extension preserving maps, and generated by monotone finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter of a certain class of ultrafilters is actually canonical when restricted to some filter base. These theorems are then applied to find connections between Tukey, Rudin-Keisler, and Rudin-Blass reducibilities on large classes of ultrafilters. The main theorems are the following. The property of having continuous Tukey reductions is inherited under Tukey reducibility, under a mild assumption. It follows that every ultrafilter Tukey reducible to a p-point has continuous Tukey reductions. If U is a Fubini iterate of p-points, then each monotone cofinal map from U to some other ultrafilter is generated (on a cofinal subset of U) by a finitary map on the base tree for U which is monotone and end-extension preserving - the analogue of continuous in this context. This is applied to prove that every ultrafilter which is Tukey reducible to some Fubini iterate of p-points has finitely generated cofinal maps. Similar theorems also hold for some other classes of ultrafilters.

K 0 of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group KO(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of KO(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding KO(R) order isomorphic to G(PrimR), where PrimR is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as KO(R) for a suitable semiartinian and unit-regular ring R.

Companions of directed sets and the Ordering Lemma

Given a partially ordered set (D,≤), a companion (C⪯) of (D,≤) is a well ordered set where C is a cofinal subsets of (D,≤) such that for every c1,c2∈C if c1≤c2 then c1⪯c2. The Ordering Lemma says that every partially ordered set has a companion. Given a directed set (D,≤) and a net f:D→X, the restriction f↾C of the net to the companion (C,⪯) of (D,≤) is a transfinite sequence. We show how the convergence and clustering of f↾C is related to the convergence and clustering of f.

On the set‐theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements

In set theory without the Axiom of Choice , we study the deductive strength of the statements (“Every partially ordered set without a maximal element has two disjoint cofinal subsets”), (“Every partially ordered set without a maximal element has a countably infinite disjoint family of cofinal subsets”), (“Every linearly ordered set without a maximum element has two disjoint cofinal subsets”), and (“Every linearly ordered set without a maximum element has a countably infinite disjoint family of cofinal subsets”). Among various results, we prove that none of the above statements is provable without using some form of choice, is equivalent to , + (Dependent Choices) implies , does not imply in (Zermelo‐Fraenkel set theory with the Axiom of Extensionality modified in order to allow the existence of atoms), does not imply in (Zermelo‐Fraenkel set theory minus ) and (hence, ) is strictly weaker than in .

On sequentially closed subsets of the real line in

We show: iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset. The statement “ is sequential” is equivalent to each one of the following propositions: Every sequentially closed subset A of includes a countable cofinal subset C, for every sequentially closed subset A of , is a meager subset of , for every sequentially closed subset A of , , every sequentially closed subset of is separable, every sequentially closed subset of is Cantor complete, every complete subspace of is Cantor complete.

Transforming rectangles into squares, with applications to strong colorings

It is proved that every singular cardinal λ admits a function rts:[λ+]2→[λ+]2 that transforms rectangles into squares. Namely, for every cofinal subsets A,B of λ+, there exists a cofinal subset C⊆λ+, such that rts[A⊛B]⊇C⊛C.As a corollary, we get that for every uncountable cardinal λ, the classical negative partition relation λ+⁄→[λ+]λ+2 coincides with the following syntactically stronger statement. There exists a function f:[λ+]2→λ+ such that for every positive integer n, every family A⊆[λ+]n of size λ+ of mutually disjoint sets, and every coloring d:n×n→λ+, there exist a,b∈A with max(a)<min(b) such that c(ai,bj)=d(i,j) for all i,j<n.

Hechler's Theorem for tall analytic P-ideals

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, ≤).

Ultimately Increasing Functions

A function $g$ between directed sets $\langle \Sigma, \succeq' \rangle$ and $\langle \Lambda, \succeq \rangle$ is called \emph{ultimately increasing} if for each $\sigma_1 \in \Sigma$ there exists $\sigma_2 \succeq' \sigma_1$ such that $\sigma \succeq' \sigma_2 \Rightarrow g(\sigma)\succeq g(\sigma_1)$. A subnet of a net $a$ defined on $\langle \Lambda, \succeq \rangle$ \cite {Ke} is nothing but a composition of the form $a \circ g$ where $g$ is ultimately increasing and $g(\Sigma)$ is a cofinal subset of $\Lambda$. While even for linearly ordered sets, an increasing net defined on a cofinal subset of the domain need not have an increasing extension, in complete generality, it must have an ultimately increasing extension, and conversely when the domain is linearly ordered. Applications are given in the context of functions with values in a linearly ordered set equipped with the order topology - in particular, the extended real numbers. For example, we show that a real sequence $\langle a_n \rangle$ converges to the supremum of its set of terms if and only if $\langle a_n \rangle$ is the supremum of the ultimately increasing sequences that it majorizes.

Coordinatization of lattices by regular rings without unit and Banaschewski functions

A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. Every (not necessarily unital) countable von Neumann regular ring R has a map \({\varepsilon}\) from R to the idempotents of R such that \({x{R} = \varepsilon(x){R}}\) and \({\varepsilon(xy) = \varepsilon(x)\varepsilon(xy)\varepsilon(x)}\) for all \({x, y \in R}\). Every sectionally complemented modular lattice with a Banaschewski trace (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice. This applies, in particular, to any sectionally complemented modular lattice with a countable cofinal subset.

A note on almost disjoint families

We give a short proof of the existence of arbitrarily large chro matic almost disjoint systems. With the same method we solve a problem of P. Szeptycki. One of the questions posed by P. Erdos and A. Hajnal in [2] was, do there exist arbitrarily large chromatic almost disjoint systems of countable sets? This was eventually proved by G. Elekes and G. Hoffmann in [1]. Later the result was extended to systems of sets of larger cardinality ([3]). These proofs applied a Baire category type argument in a topological setting. Here we give a short, direct proof of the Elekes-Hoffmann-Komjath theorem. With the same method we settle a problem of P. Szeptycki ([4]). He asked if it is consistent that for every family K of almost disjoint countable subsets of R there are sets Bo, B1,.... C R such that every H e KH has 1 ,> are infinite cardinals, then there exists an almost disjoint system Kt C [S] for some ISI = 2' with Chr (K) > i'. Proof. Set '1 = {f: a -+ r, a < s,+}. Clearly I (D = 2'. For some of the functions f E 1i we define H(f) c [4?] as follows. Let f: a -K i. If it is possible to find a set X C a of order type ,t, cofinal in a (and so cf(a) = cf(,u)) such that f is constant on X, then set f E 4I* and let H(f) = {ff13: /3 E X} for one such X. If no such set X can be found, make f V 4D* and leave H(f) undefined. Our system is K = {H(f): f E Lemma 1. KH is almost disjoint. Proof. Assume that IH(f ) n H(g)i = p Then H(f) = {f y: y E X} and H(g) = {gi6: 6 E Y} where f: a -* K, g 3 -s, X C a, Y C: are cofinal subsets of Received by the editors November 20, 2007. 2000 Mathematics Subject Classification. Primary 03E05.

A Proof of Shelah's Strong Covering Theorem for P&lt;sub&gt;k&lt;/sub&gt;&lt;sup&gt;λ&lt;/sup&gt;

Suppose that κ is a regular uncountable cardinal and λ is a cardinal > κ. We give a direct proof of Shelah’s theorem that a cofinal subset of Pκλ can be covered by some stationary set of the same size.

On the Existence of Relative Injective Covers

Let \(\sigma = ({\mathcal{T}},{\mathcal{F}})\) be a hereditary torsion theory for the category \({\mathcal{R}}\)-mod of unital left \({\mathcal{R}}\)-modules over an associative ring \({\mathcal{R}}\) with an identity element. The purpose of this note is to prove that if the associated Gabriel filter \({\mathcal{L}}\) consists of finitely presented left ideals, then every module has a \(\sigma \)-injective cover and if \({\mathcal{L}}\) contains a cofinal subset of finitely presented left ideals, then every module has a \(\sigma \)-torsionfree \(\sigma \)-injective cover. The methods used working with pure submodules contained in ``large" submodules also allow to unify the proofs of some previously known results.

Generic expansions of ω-categorical structures and semantics of generalized quantifiers

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks.

A Uniqueness Problem in Valued function Fields of Conics

Let v0 be a valuation of a field K0 with value group G0. Let K be a function field of a conic over K0, and let v be an extension of v0 to K with value group G such that G/G0 is not a torsion group. Suppose that either (K0, v0) is henselian or v0 is of rank 1, the algebraic closure of K0 in K is a purely inseparable extension of K0, and G0 is a cofinal subset of G. In this paper, it is proved that there exists an explicitly constructible element t in K, with v(t) non-torsion modulo G0 such that the valuation of K0(t), obtained by restricting v, has a unique extension to K. This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].

A uniqueness problem in simple transcendental extensions of valued fields

Let υ0 be a valuation of a field K0 with value group G0 and υ be an extension of υ0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t∈K0(x)/K0 with υ(t) non-torsion mod G0 such that υ is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is “yes” if υ0 is henselian or if υ0 is of rank 1 with G0 a cofinal subset of the value group of υ in the latter case, and that it is “no” in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, υ)/(K0, υ0).