8,035 research outputs found
Stone pseudovarieties
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
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
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
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
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 and a totally ordered
algebra , we introduce the concept of a finitely valued normal function
. We show that the operations of lift to the set of all
finitely valued normal functions, and that there is a canonical proximity
relation on . 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 are
axiomatized as proximity Baer Specker -algebras, those pairs ,
where is a torsion-free -algebra generated by its idempotents that is a
Baer ring, and is a proximity relation on . We introduce the
category of proximity Baer Specker -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 -algebras.Comment: 34 page
Idempotent generated algebras and Boolean powers of commutative rings
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
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
-subordination algebras were recently introduced as a
generalization of de Vries algebras, and it was proved that the category
of -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 -subordination algebras, and
utilize the relational nature of the morphisms in to prove
that the MacNeille completion functor establishes an equivalence between
and its full subcategory consisting of de Vries algebras. We
also generalize ideal completions of boolean algebras to the setting of
-subordination algebras and prove that the ideal completion
functor establishes a dual equivalence between 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 corresponding
to continuous relations and continuous functions between compact Hausdorff
spaces
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