Some questions concerning the cofinality of Sym(κ)

1995 ◽  
Vol 60 (3) ◽  
pp. 892-897 ◽  
Author(s):  
James D. Sharp ◽  
Simon Thomas
Keyword(s):  
Analogous Result ◽  
Infinite Cardinal ◽  
Finitely Generated ◽  
Regular Cardinals ◽  
Uncountable Cardinals ◽  
A Chain

Suppose that G is a group that is not finitely generated. Then the cofinality of G, written c(G), is denned to be the least cardinal λ such that G can be expressed as the union of a chain of λ proper subgroups. If κ is an infinite cardinal, then Sym(κ) denotes the group of all permutations of the set κ = {α∣α < κ}. In [1], Macpherson and Neumann proved that c(Sym(κ)) > κ for all infinite cardinals κ. In [4], we proved that it is consistent that c(Sym(ω)) and 2ω can be any two prescribed regular cardinals, subject only to the obvious requirement that c(Sym(ω)) ≤ 2ω. Our first result in this paper is the analogous result for regular uncountable cardinals κ.Theorem 1.1. Let V ⊨ GCH. Let κ, θ, λ ∈ V be cardinals such that(i) κ and θ are regular uncountable, and(ii) κ < θ ≤ cf(λ).Then there exists a notion of forcing ℙ, which preserves cofinalities and cardinalities, such that if G is ℙ-generic then V[G] ⊨ c(Sym(κ)) = θ ≤ λ = 2κ.Theorem 1.1 will be proved in §2. Our proof is based on a very powerful uniformization principle, which was shown to be consistent for regular uncountable cardinals in [2].

scholarly journals The cofinality spectrum of the infinite symmetric group

1997 ◽  
Vol 62 (3) ◽  
pp. 902-916 ◽  
Author(s):  
Saharon Shelah ◽  
Simon Thomas
Keyword(s):  
Symmetric Group ◽  
Inaccessible Cardinal ◽  
Singular Cardinal ◽  
Natural Numbers ◽  
Maximum Element ◽  
Infinite Symmetric Group ◽  
Regular Cardinals ◽  
Uncountable Cardinals ◽  
A Chain

AbstractLet S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem.is the main result of this paper.Theorem. Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a) C contains a maximum element.(b) If μ is an inaccessible cardinal such thatμ = sup(C ∩ μ), thenμ ∈ C.(c) if μ is a singular cardinal such thatμ = sup(C ∩ μ), thenμ+ ∈ C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.

QUASI-ISOMETRIES AND AMALGAMATIONS OF TAME COMBABLE GROUPS

1995 ◽  
Vol 05 (02) ◽  
pp. 199-204 ◽  
Author(s):  
STEPHEN G. BRICK
Keyword(s):  
Analogous Result ◽  
Finitely Generated ◽  
Hnn Extensions ◽  
One Relator Groups

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A *C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

scholarly journals What makes a complex a virtual resolution?

2021 ◽  
Vol 8 (28) ◽  
pp. 885-898
Author(s):  
Michael Loper
Keyword(s):  
Classical Theory ◽  
Toric Variety ◽  
Chain Complex ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Content Type ◽  
Graded Modules ◽  
Cox Ring ◽  
A Chain ◽  
Free Modules

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

-definability at uncountable regular cardinals

2012 ◽  
Vol 77 (3) ◽  
pp. 1011-1046 ◽  
Author(s):  
Philipp Lücke
Keyword(s):  
Partial Order ◽  
Regular Cardinal ◽  
Short Proof ◽  
Chain Condition ◽  
Generic Extension ◽  
Infinite Cardinal ◽  
Ground Model ◽  
Generic Absoluteness ◽  
Regular Cardinals ◽  
Quasi Order

AbstractLet κ be an infinite cardinal. A subset of (κκ)n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κκ)n+1 of height κ. We define and -subsets of (κκ)n as usual.Given an uncountable regular cardinal κ with κ = κ<κ and an arbitrary subset A of κκ, we show that there is a <κ-closed forcing ℙ that satisfies the κ+-chain condition and forces A to be a -subset of κκ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof.(i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L.(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a -subset of κκ × κκ.(iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.(iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κκ) under <κ-closed forcings that satisfy the κ+-chain condition is inconsistent.In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.

scholarly journals An order property for families of sets

1988 ◽  
Vol 44 (3) ◽  
pp. 294-310
Author(s):  
N. H. Williams
Keyword(s):  
Pairwise Disjoint ◽  
Chain Condition ◽  
Infinite Cardinal ◽  
Order Property ◽  
Disjoint Family ◽  
Combinatorial Properties ◽  
The Family ◽  
Almost Disjoint ◽  
A Chain ◽  
Almost Disjoint Family

AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

scholarly journals A MEALY MACHINE WITH POLYNOMIAL GROWTH OF IRRATIONAL DEGREE

2008 ◽  
Vol 18 (01) ◽  
pp. 59-82 ◽  
Author(s):  
LAURENT BARTHOLDI ◽  
ILLYA I. REZNYKOV
Keyword(s):  
Polynomial Growth ◽  
Growth Function ◽  
Finitely Generated ◽  
Mealy Machine ◽  
A Chain

We consider a very simple Mealy machine (two nontrivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like ℓα, for [Formula: see text]; that the semigroup satisfies the identity g6 = g4; and that its lattice of two-sided ideals is a chain.

scholarly journals ON THE SPLITTING NUMBER AT REGULAR CARDINALS

2015 ◽  
Vol 80 (4) ◽  
pp. 1348-1360 ◽  
Author(s):  
OMER BEN-NERIA ◽  
MOTI GITIK
Keyword(s):  
Core Model ◽  
Generic Extension ◽  
Singular Cardinal ◽  
Ground Model ◽  
The Core ◽  
Regular Cardinals ◽  
Splitting Number ◽  
Uncountable Cardinals

AbstractLetκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs(κ) = λ starting from a ground model in whicho(κ) = λ and prove that assuming ¬0¶,s(κ) = λ implies thato(κ) ≥ λ in the core model.

Admissible ordinals and intrinsic consistency

1970 ◽  
Vol 35 (3) ◽  
pp. 389-400 ◽  
Author(s):  
Michael Machtey
Keyword(s):  
Analogous Result ◽  
Fundamental Problem ◽  
Recursion Theory ◽  
Infinite Cardinal ◽  
Reduction Procedure ◽  
Partial Recursive Function ◽  
Ordinal Numbers ◽  
Partial Recursive Functions ◽  
Uniform Manner ◽  
Admissible Ordinals

This paper deals with a fundamental problem in the theory of recursion on initial segments of the ordinal numbers which was originated by Kripke [4] and Platek [8]. Following Kreisel-Sacks [3], we define a notion of relative recursiveness for this abstract recursion theory which allows us to speak of a function ƒ being weakly recursive in a set A via a partial recursive function g; this makes it reasonable to speak of partial recursive functions as ‘reduction procedures’. We say that a reduction procedure g is intrinsically consistent if for any set A there is a partial function ƒ which is weakly recursive in A via g. It is an elementary result of ordinary recursion theory that an arbitrary reduction procedure can be replaced in a uniform manner by an intrinsically consistent reduction procedure without the loss of any function f being reducible to any set A. In §3 we show that for recursion theory on the ordinals less than any infinite cardinal the analogous result is consistent with the axioms of set theory. In §5 we show that for all other cases of abstract recursion theory the analogous result fails in a very strong way to be true.

A Note on Group Rings of Certain Torsion-Free Groups

1972 ◽  
Vol 15 (3) ◽  
pp. 441-445 ◽  
Author(s):  
R. G. Burns ◽  
V. W. D. Hale
Keyword(s):  
Group Ring ◽  
Analogous Result ◽  
Free Groups ◽  
Group Rings ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Epimorphic Image ◽  
Torsion Free

AbstractAs a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

Observation of Atom Rows in Thorium Pyromellitate Using a Field Emission STEM

1977 ◽  
Vol 35 ◽  
pp. 80-81
Author(s):  
H. Todokoro ◽  
S. Nomura ◽  
T. Komoda
Keyword(s):  
Field Emission ◽  
Amorphous Carbon ◽  
Carbon Film ◽  
Dark Field Image ◽  
Dark Field ◽  
Amorphous Carbon Film ◽  
Atomic Size ◽  
Wide Range ◽  
A Chain

It is interesting to observe polymers at atomic size resolution. Some works have been reported for thorium pyromellitate by using a STEM (1), or a CTEM (2,3). The results showed that this polymer forms a chain in which thorium atoms are arranged. However, the distance between adjacent thorium atoms varies over a wide range (0.4-1.3nm) according to the different authors.The present authors have also observed thorium pyromellitate specimens by means of a field emission STEM, described in reference 4. The specimen was prepared by placing a drop of thorium pyromellitate in 10-3 CH3OH solution onto an amorphous carbon film about 2nm thick. The dark field image is shown in Fig. 1A. Thorium atoms are clearly observed as regular atom rows having a spacing of 0.85nm. This lattice gradually deteriorated by successive observations. The image changed to granular structures, as shown in Fig. 1B, which was taken after four scanning frames.

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