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 Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

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).

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.

Undergraduate students' ownership of musical learning: obstacles and options in one-to-one teaching

2013 ◽  
Vol 30 (2) ◽  
pp. 277-295 ◽  
Author(s):  
Karin Johansson
Keyword(s):  
Music Education ◽  
Undergraduate Students ◽  
Collaborative Work ◽  
Real Life ◽  
Collective Levels ◽  
One To One ◽  
The Individual ◽  
Academic Year ◽  
Made In

This paper describes a longitudinal, collaborative case study, made in the framework of the project Students' Ownership of Learning (SOL) during one academic year with one vocal teacher and two female students. The aim of the study was to relate the interaction between the teacher's and the students' intentions and expectations to the institutional level as well as to the rules and ‘real-life’ practice of the musical profession that the students are trained for. In the study, one-to-one tuition in higher music education was theorised as a culturally and historically grounded activity system consisting of relationships between musicians, instruments, music-making traditions and audiences. The concept of contradiction was used as a tool when analysing individually experienced obstacles for musical learning. The results describe how learning obstacles such as conflicting views on the purpose of the activity may be articulated, confronted and transformed into options through collaborative work. By linking the individual and collective levels of knowledge and by using professional practice as a developmental transfer, all aspects of the ‘conservatoire tradition’ may be seen as holding potential for development and expansion.

scholarly journals Ideals in bounded equality algebras

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2113-2123
Author(s):  
Akbar Paad
Keyword(s):  
Boolean Algebra ◽  
Congruence Relation ◽  
Isomorphism Theorem ◽  
Equality Algebra ◽  
One To One

In this paper, the concept of ideal in bounded equality algebras is introduced. With respect to this concepts, some related results are given. In particular, we prove that there is an one-to-one corresponding between congruence relation on an involutive equality algebra and the set of ideals on it. Also, we prove the first isomorphism theorem on equality algebras. Moreover, the notions of prime and Boolean ideals in equality algebras are introduced. Finally, we prove that ideal I of involutive prelinear equality algebra E is a Boolean ideal if and only if E/I is a Boolean algebra.

scholarly journals On the structure of Stone lattices

1970 ◽  
Vol 2 (3) ◽  
pp. 401-413 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Subdirect Product ◽  
Stone Lattice ◽  
One To One

C.C. Chen and G. Grätzer have shown that a Stone lattice is determined by a triple (C, D, ø) where C is a boolean algebra, D is a distributive lattice with 1 and ø is an e-homomorphism from C into D(D), the lattice of dual ideals of D.It is shown here that any Stone lattice is, up to an isomorphism, a subdirect product of its centre C(L) and a special Stone lattice M(L). Special Stone lattices are characterised, in the terminology of the Chen-Grätzer triple, by the fact that the e-homomorphism Φ is one to one.In this paper we characterise a special Stone lattice L as a triple (H, C, Do) where H is a distributive lattice with 0 and 1, C is a boolean e-subalgebra of the centre of H and Do is a sublattice of H with o such that d ∈ Do & c ∈ C = d ∧ c ∈ Do, and which separates the elements of C in the sense that for any c1 ≠c2 in C there is a d in Do with d ≤ c1 but d ≰ C2. It then turns out that C is C(L) and Do is the dual of D(L).

An introduction to ω-extensions of ω-groups

1982 ◽  
Vol 47 (1) ◽  
pp. 27-36 ◽  
Author(s):  
C. H. Applebaum
Keyword(s):  
Partial Answer ◽  
Inclusion Relation ◽  
Equivalence Type ◽  
One To One

Let ε stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ε(sets), Λ for the set of isols, ΛR for the set of regressive isols, and for the set of mappings from a subset of ε into ε (functions). If ƒ is a function we write δƒ and ρƒ for its domain and range, respectively. We denote the inclusion relation by ⊃ and proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β], if δƒ = α and ρƒ = β for some function ƒ with a one-to-one partial recursive extension. We denote the recursive equivalence type of a set α, {σ ∈ V ∣ σ ≃ α} by Req(α). The reader is assumed to be familiar with the contents of [1], [2], [3], and [6].The concept of an ω-group was introduced in [6], and that of an ω-homomorphism in [1]. However, except for a few examples, very little is known about the structure of ω-groups. If G is an ω-group and Π is an ω-homomorphism, then it follows that K = Ker Π and H = Π(G) are ω-groups. The question arises that if we know the structure of K and H, then what can we say about the structure of G? In this paper we will begin the study of ω-extensions, which will give us a partial answer to this question.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Surface Structure in Microtomed Sections Caused by Glass and Diamond Knives

1971 ◽  
Vol 29 ◽  
pp. 456-457
Author(s):  
J. Temple Black ◽  
William G. Boldosser
Keyword(s):  
Plastic Deformation ◽  
Epoxy Resin ◽  
Carbon Coating ◽  
Surface Damage ◽  
Body Angle ◽  
Normal Manner ◽  
Bottom Side ◽  
Cutting Direction ◽  
Made In ◽  
Diamond Knife

Ultramicrotomy produces plastic deformation in the surfaces of microtomed TEM specimens which can not generally be observed unless special preparations are made. In this study, a typical biological composite of tissue (infundibular thoracic attachment) infiltrated in the normal manner with an embedding epoxy resin (Epon 812 in a 60/40 mixture) was microtomed with glass and diamond knives, both with 45 degree body angle. Sectioning was done in Portor Blum Mt-2 and Mt-1 microtomes. Sections were collected on formvar coated grids so that both the top side and the bottom side of the sections could be examined. Sections were then placed in a vacuum evaporator and self-shadowed with carbon. Some were chromium shadowed at a 30 degree angle. The sections were then examined in a Phillips 300 TEM at 60kv.Carbon coating (C) or carbon coating with chrom shadowing (C-Ch) makes in effect, single stage replicas of the surfaces of the sections and thus allows the damage in the surfaces to be observable in the TEM. Figure 1 (see key to figures) shows the bottom side of a diamond knife section, carbon self-shadowed and chrom shadowed perpendicular to the cutting direction. Very fine knife marks and surface damage can be observed.

Iron storage sites in the myocardium as revealed by cytohistochemistry

1988 ◽  
Vol 46 ◽  
pp. 394-395
Author(s):  
M. Ashraf ◽  
F. Thompson ◽  
S. Miki ◽  
P. Srivastava
Keyword(s):  
Cardiac Tissue ◽  
Coronary Perfusion ◽  
Intracellular Iron ◽  
Ether Anesthesia ◽  
Intense Staining ◽  
Iron Storage ◽  
Thin Sections ◽  
Rat Hearts ◽  
Cacodylate Buffer ◽  
Made In

Iron is believed to play an important role in the pathogenesis of ischemic injury. However, the sources of intracellular iron in myocytes are not yet defined. In this study we have attempted to localize iron at various cellular sites of the cardiac tissue with the ferrocyanide technique.Rat hearts were excised under ether anesthesia. They were fixed with coronary perfusion with 3% buffered glutaraldehyde made in 0.1 M cacodylate buffer pH 7.3. Sections, 60 μm in thickness, were cut on a vibratome and were incubated in the medium containing 500 mg of potassium ferrocyanide in 49.5 ml H2O and 0.5 ml concentrated HC1 for 30 minutes at room temperature. Following rinses in the buffer, tissues were dehydrated in ethanol and embedded in Spurr medium.The examination of thin sections revealed intense staining or reaction product in peroxisomes (Fig. 1).

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