The depth of ultraproducts of Boolean Algebras
英文摘要:
We show in ZFC, that the depth of ultraproducts of Boolean Algebras may be bigger than the ultraproduct of the depth of those Boolean Algebras.
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。
Saharon Shelah
We show in ZFC, that the depth of ultraproducts of Boolean Algebras may be bigger than the ultraproduct of the depth of those Boolean Algebras.
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
We prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.
We prove the consistency of irr(prod limits_{i<kappa}B_i/D)< prod limits_{i<kappa}irr(B_i)/D, where D is an ultrafilter on kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk's list of problems, that is number 35. The solution applies to many other properties, e.g., Souslinity. Next, we get similar results with kappa = aleph_1 (easily we cannot have it for kappa=aleph_0) and Boolean Algebras B_i (i< kappa) of cardinality < beth_{omega_1}.
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.
Length and ultraproducts
本文在ZFC中构造了布尔代数序列,证明了积的长度严格小于积代数的长度。
Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence <B_i:i<kappa> of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i<kappa}B_i/D)<|prod_{i<kappa}C(B_i)/D|. This answers a number of problems posed by Monk.
We show that the Depth^+ of an ultraproduct of Boolean Algebras, can not jump over the Depth^+ of every component by more than one cardinality. We can have, consequently, similar results for the Depth invariant.
We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.
Canonical extensions and ultraproducts of polarities
本文研究基于格的代数的典范扩张理论,证明代数簇的典范扩张不封闭性可由特定自由代数见证,且超积封闭的极性类的稳定集格生成典范封闭的代数簇,推广了此前相关结果。
We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.
We investigate sigma-entangled linear orders and narrowness of Boolean algebras. We show existence of sigma-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.
Monk asks (problems 13, 15 in his list; pi is the algebraic density):''For a Boolean algebra B, aleph_0 <= theta <= pi (B), does B have a subalgebra B' with pi (B')= theta ?'' If theta is regular the answer is easily positive, we show that in general it may be negative, but for quite many singular cardinals - it is positive; the theorems are quite complementary. Next we deal with pi-chi and we show that the pi-chi of an ultraproduct of Boolean algebras is not necessarily the ultraproduct of the pi-chi 's. We also prove that for infinite Boolean algebras A_i (i< kappa) and a non-principal ultra…
The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i<delta >, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product prod_{i< delta} Dom(I_i), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B has free caliber lambda (i.e. if X subseteq B and |X|= lambda, then for some Y subseteq X with |Y|= lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product o…
We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal lambda with the large cardinal property, if tightness^+(B) >= lambda^+, then depth (B) >= lambda. This improves a theorem of Dow and Monk.
Let inv denote the cardinal invariants Depth^+ and Length^+ on Boolean algebras. For many singular cardinals we create a strict inequality between the product of the inv values and the inv of the product algebra. The proof holds in ZFC.
Quantifier-free formulas and quantifier alternation depth in doctrines
本文在布尔学说的代数框架中引入无量词片段和QA分层布尔学说,分别公理化模理论的无量词公式集合与按量词交替深度的公式分层,并建立了二者与带量词布尔学说之间的对应关系。
We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than kappa. We give a condition (for lambda, mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda, mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth (B_1 oplus_{A_theta} B_1) >= lambda. We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary condi…
Freely adding one layer of quantifiers to a Boolean doctrine
该研究在学说框架下刻画了对应模型全称有效公式的无量词类,构建了满足泛性质的自由QA单步布尔学说,给出了量词交替深度至多1公式的Herbrand定理的学说版本。
The canonical pair of a proof system $P$ is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have $P$-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth~$d$ Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number~$k$, also called the depth. We show that the canonical pair of a depth~$d$ Frege system is polynomially equivalent to the pair $(A_{d+2},B_{d+2})$ where $A_{d+2}$ (respectively, …
We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove consistency of the statement ``there is a Boolean algebra B such that irr(B)<s(B otimes B)'' and we force a superatomic Boolean algebra B_* such that s(B_*)=inc(B_*)=kappa, irr(B_*)=Id(B_*)=kappa^+ and Sub(B_*)=2^(kappa^+). Next we force a superatomic algebra B_0 such that irr(B_0)<inc(B_0) and a superatomic algebra B_1 such that t(B_1)>Aut(B_1). Finally we show that …
We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation. As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically v…
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