We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a ``black box'' at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.
Oscillating subalgebras of the atomless countable Boolean algebra
University of Florida（佛罗里达大学）；Charles University（查理大学）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；University of Waterloo（滑铁卢大学）；TU Dresden（德累斯顿工业大学）；University of Toronto（多伦多大学）；Institut de Mathématiques de Jussieu, CNRS（巴黎第七大学/法国国家科学研究中心儒勒·让塞数学研究所）；Matematički Institut, SANU（塞尔维亚科学与艺术学院数学研究所）
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…
We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak A$ which is not in SPM but every subalgebra of $\mathfrak A$ of cardinality $\mathfrak c$ admits a strictly positive measure. This result is essentially due to Farah and Velickovic.
Moisil logic, having as algebraic counterpart Łukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of Łukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued Łukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a fa…
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 characterise piecewise Boolean domains, that is, those domains that arise as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent descriptions of the category of piecewise Boolean algebras: either as piecewise Boolean domains equipped with an orientation, or as full structure sheaves on piecewise Boolean domains.
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.
If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{…
The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphis…
We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.
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.
An algebra $\mathbb A$ is called a perfect extension of its subalgebra $\mathbb B$ if every congruence of $\mathbb B$ has a unique extension to $\mathbb A$. This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$. Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mat…
Inspired by the fundamental results obtained by P. Halmos and A. Monteiro, concerning equivalence relations and monadic Boolean algebras, we recall the `concrete' Rauszer Boolean algebra pointed out by C. Rauszer (1971), via un preorder R. On this algebra we can consider one of the several binary operations defined, in an abstract way, by A. Monteiro (1971). The Heyting-Brouwer subalgebra of constants (fixpoints), allows us to give a general framework to find representations of several special algebraic structures related to logic.
A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a review of some generalizations of results from descriptive set theory concerning Baire category which ar…
We prove in ZFC that for mu >= aleph_2 there is a sigma --ideal I on mu and a Boolean sigma --subalgebra B of the family of subsets of mu which includes I such that the natural homomorphism from B onto B/I cannot be lifted.
We give a unified treatment of the model theory of various enrichments of infinite atomic Boolean algebras, with special attention to quantifier-eliminations, complete axiomatizations and decidability. A classical example is the enrichment by a predicate for the ideal of finite sets, and a novel one involves predicates giving congruence conditions on the cardinality of finite sets. We focus on three examples, and classify them by expressive power.
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…
It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+. In th…
We study (von Neumann) regular $^*$-subalgebras of $B(H)$, which we call R$^*$-algebras. The class of R$^*$-algebras coincides with that of "E$^*$-algebras that are pre-C$^*$-algebras" in the sense of Z. Szűcs and B. Takács. We give examples, properties and questions of R$^*$-algebras. We observe that the class of unital commutative R$^*$-algebras has a canonical one-to-one correspondence with the class of Boolean algebras. This motivates the study of R$^*$-algebras as that of noncommutative Boolean algebras. We explain that seemingly unrelated topics of functional analysis, like AF C$^*$-alge…
Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardi…
We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -i…
We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ^+,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ,μ)-FN for λ>μform a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (κ,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in t…
Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. However, it is proven that a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furt…
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…
Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances i…
It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. Our construction over the 2-element field is related to the Reed--Muller codes.
