摘要:

AbstractOn countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well‐known non‐appropriateness of the representation to some basebby proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numb

ISSN:0942-5616    

DOI:10.1002/malq.19990450202

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractWe discuss a problem asked by E. van Douwen and A. Miller [5] in various forcing models.

ISSN:0942-5616    

DOI:10.1002/malq.19990450203

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractWe construct a recursive ultrapowerF/Usuch thatF/Uis a tame 1‐model in the sense of [6, §3] andFUis existentially incomplete in the models of II2arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some do

ISSN:0942-5616    

DOI:10.1002/malq.19990450204

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractIt is shown, within constructive mathematics, that the unit ball B1(H) of the set of bounded operators on a Hilbert spaceHis weak‐operator totally bounded. This result is then used to prove that the weak‐operator continuity of the mappingT→ATonB1(H) is equivalent to the existence of the adjoin

ISSN:0942-5616    

DOI:10.1002/malq.19990450205

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractIn this note we introduce and study algebras (L, V, Λ, ⌝, 0,1) of type (2, 2,1,1,1) such that (L, V, ⌝, 0,1) is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ (a V b) = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi‐

ISSN:0942-5616    

DOI:10.1002/malq.19990450206

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractIn this paper we make an attempt to study classes of models by using general logics. We do not believe thatLwwis always the best logic for analyzing a class of models. Let K be a class of models andLa logic. The main assumptions we make about K andCare that K has theL‐amalgamation property and, later in the paper, that K does not omitL‐types. We show that, if modified suitably, most of the results of stability theory hold in this context. The main difference is that existentially closed models of K play the role that arbitrary models play in traditional stability theory. We prove e. g. a structure theorem for the class of existentially closed models of K assuming that K is a trivial superstable class with n

ISSN:0942-5616    

DOI:10.1002/malq.19990450207

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractFor an infinite cardinalKa stronger version ofK‐distributivity for Boolean algebras, called k‐partition completeness, is defined and investigated (e. g. everyK‐Suslin algebra is aK‐partition complete Boolean algebra). It is shown that everyk‐partition complete Boolean algebra isK‐weakly representable, and for strongly inaccessibleKthese concepts coincide. For regularK≥u, it is proved that an atomlessK‐partition complete Boolean algebra is an updirected union of basicK‐tree algebras. Using K‐partition completeness, the concept of γ‐almost compactness is introduced for γ ≥K. For strongly inaccessibleKwe show thatKis K‐almost compact iffKis weakly compact, and ifKis 2K‐almost compact, thenKis measurable. FurtherKis strongly compact iff it is

ISSN:0942-5616    

DOI:10.1002/malq.19990450208

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

摘要:

AbstractWe study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one‐to‐one with the set Dfc of infinite developments in continued fraction (dfc). We define two extensions of Irr, one, called Dfc1, is the set of dfc of rationals and irrationals preserving for each rational one dfc, the other, called Dfc2, is the set of dfc of rationals and irrationals preserving for each rational its two dfc. We introduce six natural distances over Irr wich we denote by dfc0, dfc1, dfc2, d, dmirand dcut. We show that only the four distances dfco, dfc1, d and dmiramong the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens. We study further the set Dfc1in which we show that the irrationals constitute a closed subset. Finally, we make a particular study of the completion Dfc2of Dfc for the two equivalent metrics dfc2and d

ISSN:0942-5616    

DOI:10.1002/malq.19990450209

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY

ISSN:0942-5616    

DOI:10.1002/malq.19990450201

出版商:WILEY‐VCH Verlag Berlin GmbH    

年代:1999

数据来源: WILEY