An ideal characterization of Mahlo cardinals

1989 ◽  
Vol 54 (2) ◽  
pp. 467-473 ◽  
Author(s):  
Qi Feng
Keyword(s):  
Regular Cardinal ◽  
Normal Ideal ◽  
Mahlo Cardinal ◽  
Mahlo Cardinals

AbstractWe show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≧ κ and λ<κ = λ then there is a nontrivial κ-complete κ-normal fine ideal on Pκ(λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ+ and μ is a regular cardinal then (1) μ< μ = μ if and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if μ< μ = μ < λ<μ = λ then there is a nontrivial κ-complete weakly κ-normal fine ideal on Pκ(λ).

A Technique to Generate -Ary Free Lattices from Finitary Ones

1985 ◽  
Vol 37 (2) ◽  
pp. 324-336
Author(s):  
George Grätzer ◽  
David Kelly
Keyword(s):  
Regular Cardinal

Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all XL satisfying 0 < |X| < m, ∧ X and ∨ X exist.This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].The -lattice is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)In [6] we proved the theorem stated below. Our proof was based on characterization of (the free -lattice on P) due to [1]; as a result, our proof was very computational.

Properties of subtle cardinals

1987 ◽  
Vol 52 (4) ◽  
pp. 1005-1019 ◽  
Author(s):  
Claudia Henrion
Keyword(s):  
Large Cardinal ◽  
Normal Ideal ◽  
Mahlo Cardinal ◽  
Normal Ideals

Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that ifκis subtle then ◇κholds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem,κis ineffable if and only ifκis subtle andΠ½-indescribableandthe subtle andΠ½-indescribable ideals cohere, i.e. they generate a proper, normal ideal (which in fact turns out to be the ineffable ideal).In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.1) We relativize the following two facts about subtle cardinals:i) ifκisn-subtle then {α<κ:αis notn-subtle} isn-subtle, andii) ifκis (n+ 1)-subtle then {α<κ:αisn-subtle} is in the (n+ 1)-subtle filter to subsets ofκ:i′) ifAis ann-subtle subset ofκthen {α ϵ A:A∩αis notn-subtle} isn-subtle, andii′) ifAis an (n+ 1)-subtle subset ofκthen {α ϵ A:A∩αisn-subtle} is (n+ 1)-subtle.2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.3) In §3 we use the technique of forcing to turn a subtle cardinal into aκ-Mahlo cardinal that is no longer subtle.4) In §4 we extend the results of §3 by showing how to turn an (n+ 1)-subtle cardinal into ann-subtle cardinal that is no longer (n+ 1)-subtle.

◇ at Mahlo cardinals

2000 ◽  
Vol 65 (4) ◽  
pp. 1813-1822 ◽  
Author(s):  
Martin Zeman
Keyword(s):  
Mahlo Cardinal ◽  
Mahlo Cardinals

AbstractGiven a Mahlo cardinal k and a regular ε such that ω1 < ε < k we show that ◇k(cf = ε) holds in V provided that there are only non-stationarily many β < k with o(β) ≥ ε in K.

On splitting stationary subsets of large cardinals

1977 ◽  
Vol 42 (2) ◽  
pp. 203-214 ◽  
Author(s):  
James E. Baumgartner ◽  
Alan D. Taylor ◽  
Stanley Wagon
Keyword(s):  
Large Cardinals ◽  
Normal Ideal ◽  
Weakly Compact ◽  
Uncountable Cardinal ◽  
Stationary Set ◽  
Stationary Subsets ◽  
Open Question ◽  
Mahlo Cardinals ◽  
Normal Ideals

AbstractLet κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.

Normality and

1991 ◽  
Vol 56 (3) ◽  
pp. 1064-1067
Author(s):  
R. Zrotowski
Keyword(s):  
Boolean Algebra ◽  
Partial Answer ◽  
Normal Ideal ◽  
Mahlo Cardinal ◽  
One To One ◽  
Made In

AbstractThe main result of this paper is that if κ is not a weakly Mahlo cardinal, then the following two conditions are equivalent:1. is κ+-complete.2. is a prenormal ideal.Our result is a generalization of an announcement made in [Z]. We say that is selective iff for every -function f: κ → κ there is a set X ∈ such that f∣(κ − X) is one-to-one. Our theorem provides a positive partial answer to a question of B. Wȩglorz from [BTW, p. 90], viz.: is every selective ideal with κ+-complete, isomorphic to a normal ideal?The theorem is also true for fine ideals on [λ]<κ for any κ ≤ λ, i.e. if κ is not a weakly Mahlo cardinal then the Boolean algebra is λ+-complete iff is a prenormal ideal (in the sense of [λ/<κ).

scholarly journals The bounded proper forcing axiom

1995 ◽  
Vol 60 (1) ◽  
pp. 58-73 ◽  
Author(s):  
Martin Goldstern ◽  
Saharon Shelah
Keyword(s):  
Regular Cardinal ◽  
Consistency Strength ◽  
Mahlo Cardinal ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Directed Set ◽  
Forcing Axiom

AbstractThe bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

AEM analysis of crystallization in Ti-based amorphous alloys

1983 ◽  
Vol 41 ◽  
pp. 270-271
Author(s):  
A.R. Pelton ◽  
A.F. Marshall ◽  
Y.S. Lee
Keyword(s):  
Magnetic Properties ◽  
Amorphous Alloys ◽  
Amorphous Materials ◽  
Isothermal Annealing ◽  
Amorphous Matrix ◽  
Nucleation And Growth ◽  
Current Interest ◽  
Amorphous Films ◽  
Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

Characterization of Coherent, Ordered Precipitates in Nickel-Base Alloys

1973 ◽  
Vol 31 ◽  
pp. 144-145
Author(s):  
B. H. Kear ◽  
J. M. Oblak
Keyword(s):  
Stable Phase ◽  
Nickel Base Superalloy ◽  
Alloy 718 ◽  
Commercial Alloy ◽  
Precipitate Morphology ◽  
Continuous Precipitation ◽  
Nickel Base ◽  
Base Superalloy ◽  
Strengthening Precipitate

A nickel-base superalloy is essentially a Ni/Cr solid solution hardened by additions of Al (Ti, Nb, etc.) to precipitate a coherent, ordered phase. In most commercial alloy systems, e.g. B-1900, IN-100 and Mar-M200, the stable precipitate is Ni3 (Al,Ti) γ′, with an LI2structure. In A lloy 901 the normal precipitate is metastable Nis Ti3 γ′ ; the stable phase is a hexagonal Do2 4 structure. In Alloy 718 the strengthening precipitate is metastable γ″, which has a body-centered tetragonal D022 structure.Precipitate MorphologyIn most systems the ordered γ′ phase forms by a continuous precipitation re-action, which gives rise to a uniform intragranular dispersion of precipitate particles. For zero γ/γ′ misfit, the γ′ precipitates assume a spheroidal.

The Use of Advanced Analytical Techniques for Characterization of the Chemical, Physical, and Crystallographic Nature of a Surface

1973 ◽  
Vol 31 ◽  
pp. 132-133 ◽  
Author(s):  
R. E. Herfert
Keyword(s):  
Surface Characterization ◽  
Analytical Techniques ◽  
X Ray Diffraction ◽  
Depth Of Penetration ◽  
X Ray ◽  
The Past ◽  
Transmission Electron ◽  
Scanning Electron

Studies of the nature of a surface, either metallic or nonmetallic, in the past, have been limited to the instrumentation available for these measurements. In the past, optical microscopy, replica transmission electron microscopy, electron or X-ray diffraction and optical or X-ray spectroscopy have provided the means of surface characterization. Actually, some of these techniques are not purely surface; the depth of penetration may be a few thousands of an inch. Within the last five years, instrumentation has been made available which now makes it practical for use to study the outer few 100A of layers and characterize it completely from a chemical, physical, and crystallographic standpoint. The scanning electron microscope (SEM) provides a means of viewing the surface of a material in situ to magnifications as high as 250,000X.

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