首页/论文/math/9201250

arXiv math/92012501992-01-15math.LOmath.GN

Factor = quotient, uncountable Boolean algebras, number of endomorphism and width

Saharon Shelah

发表信息：: Math. Japon. 37 (1992), 385--400

英文摘要：

We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number of subalgebras is not smaller than the number of endomorphisms, and other related inequalities. Lastly we deal with the obtainment of the supremum of the cardinalities of sets of pairwise incomparable elements of a Boolean algebra.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Topological density of ccc Boolean algebras - every cardinality occurs.

For every uncountable cardinal mu there is a ccc Boolean algebra whose topological density is mu .

1997-09-15math.GNmath.LO

Existence of Endo-Rigid Boolean Algebras

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many such endomorphisms in some reasonable ways. We prove that in any cardinality lambda=lambda^ {aleph_0} there is a Boolean algebra with no other endomorphisms. For this we use the so called "black boxes", but in a self contained way. We comment on how necessary the restriction on the cardinal is.

2011-05-19math.LO

An isometrically universal Banach space induced by a non-universal Boolean algebra

Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$ respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $κ$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $κ$. Such a phe…

2015-05-18math.FAmath.GNmath.LO

The number of openly generated Boolean algebras

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly sigma-filtered Boolean algebras. We show that for every uncountable regular cardinal kappa there are 2^kappa pairwise non-isomorphic openly generated Boolean algebras of size kappa > aleph_1 provided there is an almost free non-free abelian group of size kappa. The openly generated Boolean algebras constructed here are almost free. Moreover, for every infinite regular cardinal kappa we construct 2^kappa pairwise non-isomorphic Boolean algebras of size kappa tha…

2007-02-21math.LO

Complete quotient Boolean algebras

For I a proper, countably complete ideal on P(X) for some set X, can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal kappa, and that I is a kappa-complete ideal on P(kappa) containing all singletons. In this paper we provide consequences from and consistency results about completeness.

1994-01-15math.GNmath.LO

Densities of ultraproducts of Boolean algebras

We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density pi(A), resp. the topological density d(A), of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

1994-04-15math.GNmath.LO

更多相关论文收起更多论文

关于不可数长度的切选博弈的决定性

On Determinacy for Cut and Choose Games of Uncountable Length

研究关于完备布尔代数的切选博弈，在适当类超紧基数一致性假设下，证明极限版本博弈对所有完备布尔代数和后继基数λ一致可决定，还表明博弈不可决定实例与其他性质的关系。

2026-07-07math.LO

Some structural similarities between uncountable sets, powersets and the universe

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(κ)$, for a regular cardinal $κ$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[κ]^{<κ}$ for ${\cal P}(κ)$, and the ideal of sets $V$ for ${\cal C}(V)$.

2023-03-23math.LO

Independent families in Boolean algebras with some separation properties

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Ban…

2012-09-02math.FAmath.GNmath.LO

Existence of endo-rigid Boolean algebras

In [Sh:89] we, answering a question of Monk, have explicated the notion of ``a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it'' (see section 2 here) and proved the existence of one with character density aleph_0, assuming first diamondsuit_{aleph_1} and then only CH. The idea was that if h is an endomorphism of B, not among the ``trivial'' ones, then there are pairwise disjoint D_n in B with h(d_n) not subset d_n. Then we can, for some S subset omega, add an element x such that d <= x for n in S, x cap d_n=0 for n not in S while forbidding a solution for {y…

1992-01-15math.GNmath.LO

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…

1999-11-28math.LO

Models of PA: Standard Systems without Minimal Ultrafilters

We prove that bold N, the standard model of arithmetic, has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of bold N represented in N which is minimal (i.e. as in Rudin-Keisler order for partitions represented in N).

2018-01-14math.LO

有限单非阿贝尔Mal'cev代数的过滤布尔幂

Filtered Boolean powers of finite simple non-abelian Mal'cev algebras

University of Colorado Boulder（科罗拉多大学博尔德分校）；University of St Andrews（圣安德鲁斯大学）

本文研究有限单非阿贝尔Mal'cev代数关于可数无原子布尔代数的过滤布尔幂，给出其Fraïssé极限与自由代数同余类表示，并证明其自同构群具有小指标性质、强不可数共尾性和Bergman性质。

2026-09-09math.GRmath.LOmath.RA

有界算术与集合论中的力迫：类比与差异

Forcing in bounded arithmetic and set theory: analogies and differences

Charles University, Faculty of Arts（查理大学文学院）；University of Warwick, Department of Computer Science（华威大学计算机科学系）；Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）

本文概述有界算术中的多种力迫方法，与集合论力迫对比，指出可数框架可统一解释现有方法，并探讨类比能否发现新偏序。

2026-10-20math.LO

长度小于 ω_3 的基数序列的一致性定理

A consistency theorem for cardinal sequences of length $< ω_3$

Universitat de Barcelona（巴塞罗那大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）

本文证明在特定条件下，存在一个散射布尔空间，其基数序列为给定的序列 f。

2026-09-11math.LO