8,035 research outputs found

    Stone pseudovarieties

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    Profinite algebras have been extensively considered in the literature. They are the residually finite Stone topological algebras. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. In particular, we are interested in free structures relative to Stone pseudovarieties. In the unary case, the \v{C}ech-Stone compactification of the term algebra provides a construction of the absolutely free Stone topological algebra. A natural representation in a free profinite monoid using the Polish notation turns out also to be faithful for free profinite unary algebras. Looking at Stone spaces as the dual spaces of Boolean algebras, which may be taken as subalgebras of the Boolean algebra of subsets of the algebra of terms, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which, in the case of a Stone signature, culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety. A Birkhoff type theorem for residually closed Stone pseudovarieties is also established

    Admissibility via Natural Dualities

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    It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.Comment: 22 pages; 3 figure

    Checking Admissibility Using Natural Dualities

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    This paper presents a new method for obtaining small algebras to check the admissibility-equivalently, validity in free algebras-of quasi-identities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Rothlisberger that is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras

    Choice-free Stone duality

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    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras

    De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces

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    We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space XX and a totally ordered algebra AA, we introduce the concept of a finitely valued normal function f:X→Af:X\to A. We show that the operations of AA lift to the set FN(X,A)FN(X,A) of all finitely valued normal functions, and that there is a canonical proximity relation ≺\prec on FN(X,A)FN(X,A). This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain AA are axiomatized as proximity Baer Specker AA-algebras, those pairs (S,≺)(S,\prec), where SS is a torsion-free AA-algebra generated by its idempotents that is a Baer ring, and ≺\prec is a proximity relation on SS. We introduce the category of proximity Baer Specker AA-algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker AA-algebras.Comment: 34 page

    Idempotent generated algebras and Boolean powers of commutative rings

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    A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For an indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For a totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is equivalent to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.Comment: 18 page

    MF algebras and a Bishop -Stone -Weierstrass theorem result

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    This dissertation consists of two parts. In the first part, we obtain many new results about MF algebras. First, we continue the work on D. Voiculescu\u27s topological free entropy dimension deltatop (x1, ..., xn) for an n-tuple x&ar; = (x1, ..., xn) of elements in a unital C*-algebra. We also introduce a new invariant that is a C*-algebra analog of the invariant K3 introduced for von Neumann algebras. Second, we discuss a full amalgamated free product of unital MF (and residually finite-dimensional) algebras with amalgamation over a finite-dimensional C*-subalgebra. Necessary and sufficient conditions are given in this situation. In the last chapter, we study the reduced amalgamated free products of C*-algebras, and we show that a reduced free product of two full matrix algebras amalgamated over a finite-dimensional C*-algebra is an MF algebra. In the second part of the dissertation, we give an elementary proof of the Bishop-Stone-Weierstrass theorem for all unital subalgebras of M2&parl0;C&parr0; n with respect to its pure states. We show that the pure-state Bishop hull of a unital subalgebra (not necessarily selfadjoint) of M2&parl0;C&parr0; n is equal to itself. This dissertation is partially supported by a University of New Hampshire dissertation fellowship

    Ideal and MacNeille completions of subordination algebras

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    S5\mathsf{S5}-subordination algebras were recently introduced as a generalization of de Vries algebras, and it was proved that the category SubS5S\mathsf{SubS5^S} of S5\mathsf{S5}-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of S5\mathsf{S5}-subordination algebras, and utilize the relational nature of the morphisms in SubS5S\mathsf{SubS5^S} to prove that the MacNeille completion functor establishes an equivalence between SubS5S\mathsf{SubS5^S} and its full subcategory consisting of de Vries algebras. We also generalize ideal completions of boolean algebras to the setting of S5\mathsf{S5}-subordination algebras and prove that the ideal completion functor establishes a dual equivalence between SubS5S\mathsf{SubS5^S} and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5S\mathsf{SubS5^S} corresponding to continuous relations and continuous functions between compact Hausdorff spaces
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