Some classes of projective Boolean algebras

Koppelberg, Sabine

doi:10.1007/bf01428196

Cited by 7 publications

(8 citation statements)

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“…Some of the conditions in the next proposition were proved by Koppelberg [10]; see also Balcar, Jech and Zapletal [3] or Heindorf and Shapiro [8]. For the sake of completeness we give its proof.…”

Section: Lemma 11mentioning

confidence: 80%

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Boolean algebras admitting a countable minimally acting group

Błaszczyk¹,

Kucharski²,

Turek³

2014

Open Mathematics

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The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra. MSC:54B35, 54A05, 52A01, 54D30 Regular and relatively complete subalgebrasAll Boolean algebras considered here are assumed to be infinite. Boolean algebraic notions, excluding symbols for Boolean operations, follow the Koppelberg's monograph [9]. In particular, if (B ∧ ∨ − 0 1) is a Boolean algebra, then B + = B \ {0} denotes the set of all non-zero elements of B. A set A ⊆ B is a subalgebra of the Boolean algebra B, A ≤ B for short, if 1 0 ∈ A and A is closed under Boolean operations or, equivalently, − ∈ A for all ∈ A. We shall write A ∼ = B whenever A and B are isomorphic Boolean algebras. A non-empty set X ⊆ B + is called a partition of B whenever ∧ = 0 for distinct ∈ X and B X = 1 *

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Section: Lemma 11mentioning

confidence: 80%

“…Koppelberg [10]. However, another topological result obtained by Shapiro [13] implies that every Cohen algebra has to contain a dense projective subalgebra.…”

Section: Theorem 14 (Haydon's Theorem)mentioning

confidence: 99%

Boolean algebras admitting a countable minimally acting group

Błaszczyk¹,

Kucharski²,

Turek³

2014

Open Mathematics

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“…As we will see in the following, this holds more generally (and not only for inner Horn clauses). The following should be well-known, and can be shown with the same proof as given in [Kop89] for the weaker Proposition 2.19 there. We repeat the proof here for the convenience of the reader (for definitions of the notions appearing in the proof, however, we refer to [Kop89]).…”

mentioning

confidence: 71%

“…It is well-known that every element a ‰ 0 of a Boolean algebra is contained in an ultrafilter (see e.g. Corollary 2.17 in [Kop89]). So let U be an ultrafilter of C that contains c, and let f : C Ñ t0, 1u be the characteristic function of U.…”

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confidence: 99%

Tractable Set Constraints

Bodirsky¹,

Hils²

2012

jair

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Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tractable. We introduce a large class of set CSPs that can be solved in quadratic time. Our class, which we call E I, contains all previously known tractable set CSPs, but also some new ones that are of crucial importance for example in description logics. The class of E I set constraints has an elegant universal-algebraic characterization, which we use to show that every set constraint language that properly contains all E I set constraints already has a finite sublanguage with an NP-hard constraint satisfaction problem.An extended abstract of the content of this article appeared in the proceedings of IJCAI'11 [BHK11].

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“…157, 158]. (I thank S. Koppelberg for pointing out this fact to me and for supplying me with her notes[12] from which 1 learned about the above counter-example. )Let A ~ Band D be profinite groups.…”

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confidence: 99%

On Closed Subgroups of Free Products of Profinite Groups

Haran

1987

Proceedings of the London Mathematical Society

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IntroductionIf G is a free product of a family {Ai}ieI of discrete groups then a subgroup H of G is the free product of a free group F and (Af n H), where a E ~(i), i E I, and ~(i) is a set of representatives of Ai \ G / H. This is the content of the Kurosh subgroup theorem (KST). Is a similar result true for closed subgroups of free (profinite) products of pro finite groups? (Say, with F projective instead of free.)An answer to this question requires an appropriate definition of a free product over an infinite family of groups. Such a definition has been proposed, by Gildenhuys and Ribes in [3], for groups indexed by compact topological spaces so that the factors are locally equal to each other, except for neighbourhoods of one distinguished point. In spite of the fact that the KST holds for open subgroups of such free products, this definition seems to be too restrictive: if H is a closed subgroup of the free product then the groups Af n H, with a E G, i E I, need not be 'locally equal' to each other (cf. Example 2.4).We propose a very natural generalization of the free product with finitely many factors: an inverse limit of such free products (over an inverse system with mappings that send respective factors again into factors of a free product). This, essentially, also includes the definition of [3].We do not know whether the analogue of the KST holds for open subgroups of these free products. Nevertheless, if we restrict ourselves to separable groups, we give a satisfactory account of the closed subgroups of the free products.1. The analogue of the KST does not hold, in general, for closed subgroups of free products (Example 5.5).2. We define for a pro finite group G the notion of projectivity relative to a given family I of its subgroups (Definition 4.2). We show:2a. if G is a free product of the groups in I, and H is a closed subgroup of G, then H is projective relative to {fa n HI f E I, a E G};2b. conversely, if H is separable and projective relative to ~ then H is a closed subgroup of a free product G of a family I of subgroups such that ~ = {fa n HI rEI, a E G}.3. Separable relative projective pro-p-groups are in fact free products (Corollary 9.6).Hence we can answer a question of Lubotzky [13, 2.10]: 4. The KST holds for separable closed subgroups of free pro-p-products.

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Some classes of projective Boolean algebras

Cited by 7 publications

References 1 publication

Boolean algebras admitting a countable minimally acting group

Boolean algebras admitting a countable minimally acting group

Tractable Set Constraints

On Closed Subgroups of Free Products of Profinite Groups

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