Boolean Spaces Research Articles

Topological spaces as spaces of [formula omitted]-places

We prove that the class of topological spaces which are realizable as spaces of R -places of fields is closed under finite disjoint unions, closed subsets, and direct products with Boolean spaces.

Compactable semilattices

We characterize those semilattices that give rise to Boolean spaces on their associated spaces of ultrafilters. The class of 0-disjunctive semilattices, important in the theory of congruence-free inverse semigroups, plays a distinguished role in this theory.

A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

AbstractWe prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

Boolean inner-product spaces and Boolean matrices

This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n -tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.

A consistency result on cardinal sequences of scattered Boolean spaces

We prove that if GCH holds and τ = 〈κα : α < η 〉 is a sequence of infinite cardinals such that κα ≥ |η | for each α < η, then there is a cardinal-preserving partial order that forces the existence of a scattered Boolean space whose cardinal sequence is τ. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Integral Stone Rings

It is shown that certain partially ordered rings, defined by some of the properties of the totally ordered ring of integers, are exactly the bounded Z-rings, that is, the commutative f-rings with strong singular unit. The partially ordered rings in question amount to a discrete version of the rings introduced by M.H. Stone for his abstract characterization of the rings of real-valued continuous functions on compact Hausdorff spaces, and the function rings they correspond to are given by the integer-valued continuous functions on Boolean spaces.

A Coalgebraic View of Heyting Duality

We give a coalgebraic view of the restricted Priestley duality between Heyting algebras and Heyting spaces. More precisely, we show that the category of Heyting spaces is isomorphic to a full subcategory of the category of all Γ-coalgebras, based on Boolean spaces, where Γ is the functor which maps a Boolean space to its hyperspace of nonempty closed subsets. As an appendix, we include a proof of the characterization of Heyting spaces and the morphisms between them.

Some notes concerning the homogeneity of Boolean algebras and Boolean spaces

In this article we consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated.It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a∈D the relative algebra B↾a:={b∈B:b⩽a} is isomorphic to B. In particular, the free product of countably many copies of an atomic Boolean algebra is homogeneous.Moreover, a Boolean algebra B is homogeneous if it satisfies the following conditions: (i)B has a countably generated ultrafilter,(ii)B is not c.c.c., and(iii)for every a∈B⧹{0} there are finitely many automorphisms h1,…,hn of B such that 1=h1(a)∪⋯∪hn(a).These results generalize theorems due to Motorov [Russian Math. Surveys 44 (16) (1989) 190–191] on the homogeneity of first countable Boolean spaces.Finally, we provide three constructions of first countable homogeneous Boolean spaces that are linearly ordered. The first construction gives separable spaces of any prescribed weight in the interval [ℵ0,2ℵ0]. The second construction gives spaces of any prescribed weight in the interval [ℵ1,2ℵ0] that are not c.c.c. The third construction gives a space of weight ℵ1 which is not c.c.c. and which is not a continuous image of any of the previously described examples.

Dual spaces of some congruence lattices

For any set X and any n⩾3 we define a topological space L n(X) and characterize its closed subspaces if |X|⩽ℵ 1 . As an application we obtain a characterization of congruence lattices of algebras in some varieties of lattices. The spaces L n(X) are close to Boolean spaces, but they are not Hausdorff.

Functor category dualities for varieties of Heyting algebras

Let A be a finitely generated variety of Heyting algebras and let SI( A) be the class of subdirectly irreducible algebras in A . We prove that A is dually equivalent to a category of functors from SI( A) into the category of Boolean spaces. The main tool is the theory of multisorted natural dualities.

Attainment of Tightness in Boolean Spaces

We consider some questions of Donald Monk related to attainment of the tightness function in Boolean spaces. Then we prove in ZFC that attainment of tightness in the sense of the definition does not imply attainment of tightness in the free sequence sense. Our results give rise to full answers to [2, Problems 41, 42].

π-character and depth in scattered Boolean spaces

The relationship between π-character and depth for the class of scattered Boolean spaces is studied in [A. Dow, J.D. Monk, Topology Appl. 75 (1997) 183–199]. Some more results on this relationship are then obtained here.

Natural dualities for quasivarieties generated by a finite commutative ring

Let R be a finite commutative ring with identity. If the Jacobson radical of R annihilates itself, then the quasivariety generated by R is dually equivalent to a category of structured Boolean spaces obtained in a natural way from R. If on the other hand the radical of R does not annihilate itself, then no such natural dual equivalence is possible. To illustrate the first result, a dual equivalence for the quasivariety generated by the ring \( \Bbb Z _{p^2} \), where p is prime, is given.

Quasi-modal algebras

In this paper we introduce the class of Boolean algebras with an operator between the algebra and the set of ideals of the algebra. This is a generalization of the Boolean algebras with operators. We prove that there exists a duality between these algebras and the Boolean spaces with a certain relation. We also give some applications of this duality.

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗ C^* -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

A hierarchy of maps between compacta

AbstractLet CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair (K, L) of subclasses of CH, we define Lev≥α(K, L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev≥α(BS, BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥ω a for all ordinals α: of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ ω. The results of this paper include:(i) every map of level ≥ ωis co-elementary;(ii) the limit maps of an co-indexed inverse system of maps of level ≥ α are also of level ≥ α; and(iii) if K is a co-elementary class, k &gt; ω and Lev≥k(K,K) = Lev≥k+1(K,K), then Lev≥1(K,K) = Lev≥(K,K).A space X ∈ K is co-existentially closed inK if Lev≥0(K, X) = Lev≥1(K,X). Adapting the technique of “adding roots,” by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one.

Shuffle products and other operations in the class of primitive spaces

A Boolean space is a compact Hausdorff zero-dimensional space. Boolean spaces arise as Stone duals of Boolean algebras and the duals of the countable Boolean algebras are the countably baaed (or equivalently metrizable) Boolean spaces. Among the latter spaces, primitive spaces and more particularly finitary spaces form two very peculiar subclasses. A countably baaed Boolean space is primitive if it admits a basis of pi clopen sets, where X is pi if for each clopen U C X, either X z U or X Z X \ U. It is finitary if the set of homeomorphism types of its pi clopen subsets is finite. These spaces have been introduced through utterly different means by Hanf [2], Pierce [21] and also by Paljutin 1201. They were also studied by Dobbertin [I], Heindorf [12], Myers [18], the author [6] and others ([17], [23] . . . ). Primitive spaces can be handled by easy-to-use combinatorial tools : their diagrams ([2] or [7]) and the structure of their homeomorphism types is at present fairly well understood ([22], [12], [7]). Their pl ace in Ketonen’s hierarchy ([13]) is well-known, at least in the finitary case ([12]). In the class of primitive spaces, Tarski’s cube problem (X+X+X E X implies X+X E X, see [2l] or [S]) has a positive answer, but otherwise the diagram technique enables to produce easy examples of pathological behaviour of product of spaces ([22], [S]). It should be noted that the primitive spaces bear some surprising relationships with the variable-free fragment of the modal system K4 ([9]) and that Paljutin has used finitary spaces to obtain some decidability results ([20]). Finally, it turns out that many Lindenbaum-Tarski algebras of important first-order theories (such as abelian groups, well-orders and linear orders, Boolean algebras, . . ) are primitive ([lS]). It is well-known that the class of primitive spaces is closed under the formation of finite topological sums and finite Cartesian products ([22], [24]). In this paper, we introduce other operations on Boolean spaces under which the class of primitive

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B2 of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B2. The axioms of Schmid [7], [8] for the AC and EC members of B2 are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B2 and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B2, this method transfers the entire discussion into an equivalent dual category X2 of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X2 prove to be much more readily constructed and analyzed than the corresponding algebras in B2.

A Ramsey theorem for polyadic spaces

A polyadic space is a Hausdorff continuous image of some power of the one-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collectio

Centers and Two-Sided Ideals of Right Self-Injective Regular Rings

In this paper we prove two main results. The first one is the existence of a right self-injective regular ring of type III with center any given commutative self-injective regular ring. The second one is a characterization of the lattice of two-sided ideals for a large class of right self-injective regular rings of type III. In a previous paper, this lattice was described as the lattice of order ideals of a certain lattice of continuous functions. Now we prove the converse: for a large class of complete boolean spaces X, let C be the set of all continuous functions from X into {0} ∪ [ℵ0, γ], where γ is any infinite cardinal number, and let Δ be the lattice of all functions of C that are less than or equal to a given nonvanishing function of C. Then there exists a right self-injective regular ring R of type III such that the lattice of two-sided ideals of R is isomorphic to the lattice of order ideals of Δ.