Bohuslav Balcar, Thomas Jech, Jindřich Zapletal
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A >>Cohen algebra<< is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.
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We show that there are semi-Cohen Boolean algebras which cannot be completely embedded into Cohen Boolean algebras. Using the ideas from this proof, we give a simpler argument for a theorem of S. Koppelberg and S. Shelah, stating that there are complete subalgebras of Cohen algebras which are not Cohen themselves.
We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.
It is shown that there exists a complete, atomless, sigma-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC.
The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…
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_…
We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting.
Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped with product measure, and C(kappa), the algebra of regular open sets in the space {0,1}^kappa, for kappa an infinite cardinal. C(kappa) is much easier to analyse than B(kappa) : C(kappa) has a dense subset of size kappa, while the density of B(kappa) depends on the cardinal characteristics of the real line; and the definition of C(kappa) is simpler. Indeed, C…
Almost refinement, reaping, and ultrafilter numbers
本文研究了布尔代数中最大反链的组合结构，探讨了收获关系及超滤子数的性质，并通过参数化钻石原理证明了科恩代数的超滤子数等于ℵ₁。
Under the assumption that the continuum c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(N)/fin has under CH and in the aleph2-Cohen model. We prove a similar result in the category of Banach spaces.
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean valued models and full details of all proofs are given. We focus our presentation using the boolean algebra language and defining an iteration system as a directed and commutative system of complete and injective homomorphisms between complete and atomless boolean algebras. It is well known that the boolean algebra approach to forcing and iterations is fully e…
This paper illustrates the relationship between boolean propositional algebra and semirings, presenting some results of partial ordering on boolean propositional algebras, and the necessary conditions to represent a boolean propositional subalgebra as equivalent to a corresponding boolean propositional algebra. It is also shown that the images of a homomorphic function on a boolean propositional algebra have the relationship of boolean propositional algebra and its subalgebra. The necessary and sufficient conditions for that homomorphic function to be onto-order preserving, and also an extensi…
The random ordered graph is a semi-retract of the canonically ordered atomless Boolean algebra
本文证明随机有序图是规范有序无原子布尔代数的半收缩，回答了Bartošová和Scow的公开问题。
We modify the game Fuchino, Koppelberg, and Shelah used to characterize the $κ$-Freese-Nation property for a given Boolean algebra $A$, replacing players I and II each with a team of $n$ players with limited information. We show that $A$ is tightly $κ$-filtered exactly when team II has a winning strategy for every finite team size. Case $κ=\aleph_0$ characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of very I-favorable. We similarly characterize Cohen algebras in terms of a t…
Small masas of the Calkin algebra in the Cohen model
本文在 Cohen 模型下研究 Calkin 代数的极大交换子代数，证明其密度可小于连续统，并刻画了由布尔代数构造的 masa 的同构与酉等价类型，同时给出 ZFC 中不存在密度小于 𝔭 的 masa 的结论。
On the Logical and Algebraic Aspects of Reasoning with Formal Contexts
本文研究形式背景下不同概念的表示与推理的逻辑与代数层面，提出两类模态逻辑分别处理粗糙与形式概念，统一为双类布尔模态逻辑并证明双布尔代数的刻画，探讨了系统扩展方向。
A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Will…
We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP…
In this paper we study the dense elements and the radical of a residuated lattice, residuated lattices with lifting Boolean center, simple, local, semilocal and quasi-local residuated lattices. BL-algebras have lifting Boolean center; moreover, Glivenko residuated lattices which fulfill a certain equation (that is satisfied by BL-algebras) have lifting Boolean center.
Connecting generalized Priestley duality to Hofmann-Mislove-Stralka duality
本文建立了分配格及分配交半格的Priestley对偶与半格的Hofmann-Mislove-Stralka对偶之间的联系，并通过代数框架间的态射将Stone对偶纳入统一框架。
We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with ``few automorphisms'', with entangled sequences of linear orders, and with semi-ZFC examples of the non-attainment of the spread (and hL, hd).
Ellis enveloping semigroups in real closed fields
本文引入 d-半代数集的布尔代数，证明其 Stone 空间与半代数集布尔代数的 Stone 空间的 Ellis 包络半群自然同构，并研究其与外部可定义集族的关系及表示函数。
New perspectives on semi-primal varieties
该研究运用范畴论探究半本原格扩张生成的簇，给出Keimel-Werner拓扑对偶性新证明及离散版本，分析两类簇间的伴随关系，还得到半本原簇代数典范扩张的新刻画。
Combinatorics in 1-semi-equational theories
本文通过直接组合证明，表明1-半等式理论的布尔组合具有几乎线性Zarankiewicz界，从而排除无限域的可解释性，并证明$(k,1)$-半等式满足强Erdős--Hajnal性质。
The Lifting Idempotent Property ($LIP$) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures ($MV$-algebras, commutative l-groups, $BL$-algebras, bounded distributive lattices, residuated lattices,etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property ($CBLP$) for congruence modular algebras). A lifting ideal of a ring $R$ is an ideal of $R$ fulfilling $LIP$. In a recent paper, Tarizadeh and Sharma obtained new results on liftin…
The class of involutive bisemilattices plays the role of the algebraic counterpart of paraconsistent weak Kleene logic. Involutive bisemilattices can be represented as Plonka sums of Boolean algebras, that is semilattice direct systems of Boolean algebras. In this paper we exploit the Plonka sum representation with the aim of counting, up to isomorphism, finite involutive bisemilattices whose direct system is given by totally ordered semilattices.
Contact semilattices
该研究给出带弱接触关系的并半格可嵌入带重叠/可加接触关系的布尔代数或分配格的精确条件，并证明带弱接触的对应代数结构也有类似刻画。
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