Transfinite Construction Research Articles

On bound of transfinite construction of inflationary fixed point

On bound of transfinite construction of inflationary fixed point

Retraction Note to: Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an algebraic refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

Metastability in the Furstenberg–Zimmer tower

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemeredi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the ωω ω th level.

Measure theory based on lattices and transfinite recursion

In this methodological study we develop the foundations of measure theory using lattices as prime structures instead of rings. Topological as well as abstract regularity is incorporated into this approach from the outset. The use of inner and outer measures is replaced by transfinite constructions. Basic extension steps are transfinitely iterated to yield generalizations of Caratheodory’s theorem which are optimal with respect to inner and outer approximations.

On the construction of free algebras for equational systems

The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.

Order-isomorphic [formula omitted]-orderings in Cohen extensions

In this paper we prove that, in the Cohen extension (adding ℵ 2 -generic reals) of a model M of ZFC+CH containing a simplified ( ω 1 , 1 ) -morass, η 1 -orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η 1 -orderings can be extended to an order-isomorphism between the η 1 -orderings in the Cohen extension of M . We use the simplified ( ω 1 , 1 ) -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding ℵ 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω 1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η 1 -orderings are order-isomorphic.

Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

RETRACTED ARTICLE: Understanding the Small Object Argument

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an “algebraic” refinement of the small object argument, cast in terms of Grandis and Tholen’s natural weak factorisation systems, which rectifies each of these three deficiencies.

The Locally Free Relatively Filtered Diagram as an Inductive Completion of a System of Choice

Guitart and Lair have established the existence of Locally Free Diagrams, which can be seen as a purely categorical version of the solution set condition, and of the Lowenheim–Skolem theorem. Their proof is based on a transfinite construction by saturation. An iterative principle is established, but the construction is not effective for every step. The thesis of Gerner contains a more effective proof for the existence of Locally Free Diagrams (with the restriction that the projective bases of the sketch S must all be finite). But the problem of lies in the impossibility to name concretely the elements of the Locally Free Diagrams. The present paper will provide a new construction of the Locally Free Diagram in which the effective and the non-effective part will be much more separated (again the projective bases must all be finite). This construction represents a notable improvement with regard to the proof of allowing the concrete designation of the elements ofthe Locally Free Diagrams. Furthermore we show that the construction is relatively filtered (i.e. satisfies the “filtered”-property).

An order property for families of sets

AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

Spectrum reducing extension for one operator on a Banach space

In this paper we show that, given an operator T T on a Banach space X X , there is an extension Y Y of X X such that T T extends in a natural way to an operator T ∼ {T^ \sim } on Y Y , and the spectrum of T ∼ {T^ \sim } is the approximate point spectrum of T T . This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra L L containing elements g 1 {g_1} and g 2 {g_2} such that in no extension L ′ L’ of L L are the residual spectra of g 1 {g_1} and g 2 {g_{_2}} eliminated simultaneously.

Two addenda to the author's ‘Transfinite constructions’

Since the author's article “A unified treatment of transfinite constructions …”, in Volume 22 (198O) of this Bulletin, had an encyclopaedic goal, he now takes the opportunity to answer two further questions raised since that article was submitted. The lesser of these asks whether the only pointed endofunctors for which every action is an isomorphism are the well-pointed ones, at least when the endofunctor is cocontinuous; a counter-example provides a negative answer. The more important question concerns the reflexion from the comma-category T/A into the category of algebras for the pointed endofunctor T of A, and the algebra-reflexion sequence which converges to this reflexion; and asks for simplified descriptions in the special case where T is cocontinuous. We give closed formulas in this case, both for the reflexion and for the sequence which converges to it. The reader may wonder why we care about the approximating sequence when we have a closed formula for the reflexion; the answer is that, in certain applications, we need to separate the roles of finite colimits and filtered ones.

A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake.Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/AWe show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the Tk preserve either colimits or unions of suitably-long chains of subobjects.The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.

On the Cardinality of a Topological Space

In recent papers, B. Šapirovskiĭ, R. Pol, and R. E. Hodel have used a transfinite construction technique of Šapirovskiĭ to provide a unified treatment of fundamental inequalities in the theory of cardinal functions. Šapirovskiĭ’s technique is used in this paper to establish an inequality whose countable version states that the continuum is an upper bound for the cardinality of any Lindelöf space having countable pseudocharacter and a point-continuum separating open cover. In addition, the unified treatment is extended to include a recent theorem of Šapirovskiĭ concerning the cardinality of ${T_3}$ spaces.

On the cardinality of a topological space

In recent papers, B. Šapirovskiĭ, R. Pol, and R. E. Hodel have used a transfinite construction technique of Šapirovskiĭ to provide a unified treatment of fundamental inequalities in the theory of cardinal functions. Šapirovskiĭ’s technique is used in this paper to establish an inequality whose countable version states that the continuum is an upper bound for the cardinality of any Lindelöf space having countable pseudocharacter and a point-continuum separating open cover. In addition, the unified treatment is extended to include a recent theorem of Šapirovskiĭ concerning the cardinality of T 3 {T_3} spaces.

Free Vector Lattices

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

Insufficiency of the hyperspace

I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.