Menachem Magidor, Saharon Shelah
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.
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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.
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 in ZFC, that the depth of ultraproducts of Boolean Algebras may be bigger than the ultraproduct of the depth of those 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 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.
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。
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…
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。
On the arithmetic of ultraproducts of commutative cancellative monoids
本文研究交换可消幺半群超乘积中元素的不可约分解，完整刻画了长度多重集，并利用超乘积技术证明分解理论若干性质非一阶可表达、构造实现任意长度多重集的整环，同时给出加法组合学定理的新证明。
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…
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.
We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct li…
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 …
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。
Johnson's axioms revisited: Bases for Boolean algebras containing identities of associative type. I
本文受Johnson布尔代数公理体系启发，证明其第三条公理冗余，给出多个含非结合律的结合型恒等式的布尔代数等式基，可转化为经典命题逻辑公理体系。
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to substitute lengthy logical considerations by schematic algebraic manipulations.
A consistency theorem for cardinal sequences of length $< ω_3$
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。
We consider $ω^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $ω^n$ for some integer $n\geq 1$. We show that all these structures are $ω$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $ω^2$-automatic (resp. $ω^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above…
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