On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

On the S 2-Fication of Some Toric Varieties

In this article we prove the following:

1. Some results on the Cohen–Macaulayness of the canonical module;

2. We study the S ₂-fication of rings which are quotients by lattices ideals;

3. Given a simplicial lattice ideal of codimension two I, its Macaulayfication is given explicitly from a system of generators of I.

     

THE IMPACT OF THE NATIONAL NUMERACY STRATEGY IN YEAR 4 (II): TEACHING

The Nuffield Year 4 project set out to examine selected aspects of the impact of the National Numeracy Strategy (NNS) in primary schools by using comparable data on Year 4 pupil attainment and teaching collected in 1997/98 and 2001/02, two years before and two years after the introduction of the Numeracy Strategy. In this second paper we address the following questions:

• How and to what extent has numeracy teaching in Year 4 changed?

• Why have these changes occurred?

• How can we improve future implementation of reforms?

• How can we inform continuing professional development?

     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

Products of Quasi-p-Pseudocompact Spaces

Given p () , we determine when a product of quasi-p-pseudocompact spaces preserves this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of () containing . We give examples of spaces X, Y, X _s , Ys which are quasi-p-pseudocompact for every p *, but X Y is not pseudocompact, and X _s Y _s is pseudocompact and it is not quasi-s-pseudocompact for each s *. Besides, we prove that every pseudocompact space X of with X, is quasi-p-pseudocompact for some p *. Finally, we introduce, for each p *, the class P _p of all spaces X such that X × Y is quasi-p-pseudocompact when so is Y; and we prove: (1) the intersection of classes P _p ( _p *) coincides with the Frol"ik class; (2) every class P _p is closed under arbitrary products; (3) the partial ordered set ( P _p p ,) is isomorphic to the set of equivalence classes of free ultrafilters on with the Rudin–Keisler order. A topological characterization of RK-minimal ultrafilters is also given.     

A note on weakly compact subsets in the projective tensor product of Banach spaces

Let X and Y be two Banach spaces. In this short note we show that every weakly compact subset in the projective tensor product of X and Y can be written as the intersection of finite unions of sets of the form , where K_X and K_Y are weakly compacts subsets of X and Y, respectively. If either X or Y has the Dunford–Pettis property, then any intersection of sets that are finite unions of sets of the form , where K_X and K_Y are weakly compact sets in X and Y, respectively, is weakly compact.     

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗̌_ɛ Y be the injective tensor product of X and Y.

The Morse minimal system is nearly continuously Kakutani equivalent to the binary odometer

Ergodic homeomorphisms T and S of Polish probability spaces X and Y are evenly Kakutani equivalent if there is an orbit equivalence ?: X ₀ → Y ₀ between full measure subsets of X and Y such that, for some A ? X ₀ of positive measure, ? restricts to a measurable isomorphism of the induced systems T _A and S _?(A). The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two zero-entropy loosely Bernoulli systems are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as the Morse minimal system are both measurable and topological.Recently del Junco, Rudolph and Weiss studied a new relation, called nearly continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an even Kakutani equivalence where also X ₀ and Y ₀ are invariant G _δ sets, A is within measure zero of both open and closed, and ? is a homeomorphism from X ₀ to Y ₀. It is known that nearly continuous Kakutani equivalence is strictly stronger than even Kakutani equivalence, and nearly continuous Kakutani equivalence is the natural strengthening of even Kakutani equivalence to the nearly continuous category—the category of maps that are continuous after sets of measure zero are removed. In this paper, we show that the Morse minimal substitution system is nearly continuously Kakutani equivalent to the binary odometer.     

The packing measure of a general subordinator

Summary Precise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y ₁ (t), Y ₂ (t)), whereY ₁ andY ₂ are independent copies ofX. A finite and positive packing measure is possible for subordinators close to Cauchy; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).Research supported in part by NSF under contracts (1) DMS 87-01866, and (2) DMS 87-01212     

Characterization of Abelian groups with a minimal generating set

Abstract

We characterize Abelian groups with a minimal generating set: Let τ A denote the maximal torsion subgroup of A. An infinitely generated Abelian group A of cardinality κ has a minimal generating set iff at least one of the following conditions is satisfied:

1. dim(A/pA) = dim(A/qA) = κ for at least two different primes p, q.

2. dim(t A/pt A) = κ for some prime number p.

3. Σ{dim(A/(pA + B)) ∣ dim(A/(pA + B)) < κ} = κ for every finitely generated subgroup B of A.

Moreover, if the group A is uncountable, property (3) can be simplified to (3') Σ{dim(A/pA) ∣ dim(A/pA) < κ} = κ, and if the cardinality of the group A has uncountable cofinality, then A has a minimal generating set iff any of properties (1) and (2) is satisfied.     

Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X_∞ and X_∞^d being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (A_α) α ? I of upward filtering finite σ-algebras in A. X_∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X_∞. Further on we obtain that each measurable set of the remainder of X_∞ has measure zero with respect to the limit measure P_∞ (Theorem 1). X_∞^d is the STONE representation space X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A_α, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) and x(A) where x(A) is the Stone representation space of A, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of Alexandroff Hausdorff for compact T₂ measure spaces x with regular Borel measure (Theorem 3.).     

On isomorphisms of some Köthe function <Emphasis Type="Italic">F</Emphasis>-spaces

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (Ω _X ; Σ _X , µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ), respectively, with absolute continuous norms are isomorphic and have the property

$\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$

(for µ = µ _X and µ = µ _Y , respectively) then the measure spaces (Ω _X ; Σ _X ; µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L _p (µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

     

Four-Dimensional Real Third-Power Associative Division Algebras

We study third-power associative division algebras A over a field 𝕂 of characteristic different from 2. Those algebras having dimension ≤2 are commutative. When 𝕂 is the field ? of real numbers, those algebras having dimension 4 are power-commutative in each of the following two cases:

1. A contains a central element;

2. A satisfies the additional identity (x, x³, x) = 0.

     

Between associativity and commutativity

For any binary operation, four alternatives exist. It could be

1. (i) both commutative and associative;

2. (ii) neither commutative nor associative;

3. (iii) commutative but not associative;

4. (iv) associative but not commutative.

The basic arithmetical operations provide examples for the first two possibilities. This paper presents elementary examples for the last two possibilities. It is claimed that the study of these examples is extremely important in order to understand the logical independence of commutativity and associativity.     

Holomorphic mappings of once-holed tori

Let T be the space of marked once-holed tori and Y₀ be a Riemann surface with marked handle. We investigate geometric properties of the set T_a[Y₀] of X ∈ T that allow holomorphic mappings of X into Y₀. We also examine the set T_c[Y₀] of marked once-holed tori conformally embedded into Y₀. It turns out that T_a[Y₀] and T_c[Y₀] have several properties in common. Our basic tool is a new notion, called a handle condition.     

AN AXIOMATIC APPROACH TO CATEGORIES OF TOPOLOGICAL ALGEBRAS

Abstract

Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:

1. if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;

2. V carries U-initial monosources into T-initial mono-sources.

The functor V has many nice properties which shed light on the blending of the “topology” and “algebra”; e.g., V is a topologically algebraic functor in the sense of Y.H. Hong. An ([Etilde],[Mtilde]) version of O. Wyler's “Taut Lift Theorem” is used to show that the existence of a left adjoint to V is related to Condition (ii). It is also shown that certain topological algebraic reflections arise as Topological Algebraic Situations from algebraic and topological surjective reflections.     

Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case

Summary LetX ₁,X ₂, ...,X _r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX _i =(X _i1 , ...,X _in ) and the random vectorY=(Y ₁, ...,Y _n ), their maximum, is defined byY _j =max{X _ij :1ir}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ ₁,Z ₂, ...,Z _s . It is then shown in this paper that the distributions of theZ _i 's are simply a rearrangement of those of theZ _j 's (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.