On a generalization of distributivity

1994 ◽  
Vol 59 (3) ◽  
pp. 1055-1067 ◽  
Author(s):  
Yasuo Kanai
Keyword(s):  
Prime Ideal ◽  
Simple Proof ◽  
Regular Cardinal ◽  
Large Cardinals ◽  
Prime Ideal Theorem ◽  
The Ideal

In this paper, we generalize the notion of distributivity and consider some properties of distributive ideals, that is, ideals I such that the algebra P(κ)/I is distributive in our sense.Our notation and terminology is explained in §1, while the main results of this paper begin in §2. We shall show here some relations of the distributivity and the ideal theoretic partitions. In §3, we shall study the class of distributive ideals over κ whose existence is equivalent to the ineffability of κ, and other classes. Finally, in §4, we shall consider the equivalence of the Boolean prime ideal theorem and show that the existence of certain distributive ideals characterizes several large cardinals. As a byproduct, we can give a simple proof of Ketonen's theorem that κ is strongly compact if and only if for any regular cardinal λ ≥ κ there exists a nontrivial κ-complete prime ideal over λ.

Combinatorics on large cardinals

1992 ◽  
Vol 57 (2) ◽  
pp. 617-643 ◽  
Author(s):  
Carlos H. Montenegro E.
Keyword(s):  
Regular Cardinal ◽  
Ordered Set ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Combinatorial Properties ◽  
A Chain ◽  
Linearly Ordered Set ◽  
Normal Ideals ◽  
The Ideal

Our framework is ZFC, and we view cardinals as initial ordinals. Baumgartner ([Bal] and [Ba2]) studied properties of large cardinals by considering these properties as properties of normal ideals and not as properties of cardinals alone. In this paper we study these combinatorial properties by defining operations which take as input one or more ideals and give as output an ideal associated with a large cardinal property. We consider four operations T, P, S and C on ideals of a regular cardinal κ, and study the structure of the collection of subsets they give, and the relationships between them.The operation T is defined using combinatorial properties based on trees 〈X, <T〉 on subsets X ⊆ κ (where α <T β → α < β). Given an ideal I, consider the property *: “every tree on κ with every branching set in I has a branch of size κ” (where a branching set is a maximal set with the same set of <T-predecessors, and a chain is a maximal <T-linearly ordered set; for definitions see §2). Now consider the collection T(I) of all subsets of κ that do not satisfy * (see Definition 2.2 and the introduction to §5). The operation T provides us with the large cardinal property (whether κ ∈ T(I) or not) and it also provides us with the ideal associated with this large cardinal property (namely T(I)); in general, we obtain different notions depending on the ideal I.

scholarly journals On Selberg's Lemma For Algebraic Fields

1955 ◽  
Vol 7 ◽  
pp. 138-143 ◽  
Author(s):  
R. G. Ayoub
Keyword(s):  
Prime Ideal ◽  
Simple Proof ◽  
Inversion Formula ◽  
Distribution Of Primes ◽  
Fundamental Lemma ◽  
Möbius Inversion ◽  
Möbius Inversion Formula ◽  
Algebraic Fields ◽  
Prime Ideal Theorem

1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.

Distributive ideals and partition relations

1986 ◽  
Vol 51 (3) ◽  
pp. 617-625 ◽  
Author(s):  
C. A. Johnson
Keyword(s):  
Prime Ideal ◽  
Structural Property ◽  
Regular Cardinal ◽  
Complete Ideal ◽  
Unique Member ◽  
Partition Relations ◽  
Image Position ◽  
The Ideal

It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard (see [7]) and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal of nonstationary subsets ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+(=P(κ) −I), then anI-partitionofAis a maximal collectionW⊆,P(A) ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y. (i.e.,WβrefinesWα).

Powers of the ideal of Lebesgue measure zero sets

1991 ◽  
Vol 56 (1) ◽  
pp. 103-107
Author(s):  
Maxim R. Burke
Keyword(s):  
Partial Order ◽  
Lebesgue Measure ◽  
Regular Cardinal ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Zero Sets ◽  
Covering Lemma ◽  
Inner Model ◽  
The Ideal

AbstractWe investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.

scholarly journals A note on the invertible ideal theorem

1984 ◽  
Vol 25 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Simple Proof ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Central Element ◽  
Commutative Case ◽  
Additional Information ◽  
The Subject

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

2014 ◽  
Vol 53 (7-8) ◽  
pp. 825-833 ◽  
Author(s):  
Paul Howard ◽  
Eleftherios Tachtsis
Keyword(s):  
Prime Ideal ◽  
Selection Lemma ◽  
Rado’S Selection Lemma ◽  
Prime Ideal Theorem

scholarly journals On the number of sign changes in the remainder-term of the prime-ideal theorem

1988 ◽  
Vol 56 (1) ◽  
pp. 185-197 ◽  
Author(s):  
J. Kaczorowski ◽  
W. Staś
Keyword(s):  
Prime Ideal ◽  
Remainder Term ◽  
Sign Changes ◽  
Prime Ideal Theorem

scholarly journals On the prime ideal theorem

1968 ◽  
Vol 20 (1-2) ◽  
pp. 233-247 ◽  
Author(s):  
Takayoshi MITSUI
Keyword(s):  
Prime Ideal ◽  
Prime Ideal Theorem
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