6,039 research outputs found
Completions of Leavitt path algebras
We introduce a class of topologies on the Leavitt path algebra of
a finite directed graph and decompose a graded completion
as a direct sum of minimal ideals.Comment: 16 pages and 2 figure
The periodic cyclic homology of crossed products of finite type algebras
We study the periodic cyclic homology groups of the cross-product of a finite
type algebra by a discrete group . In case is commutative and
is finite, our results are complete and given in terms of the singular
cohomology of the strata of fixed points. These groups identify our cyclic
homology groups with the \dlp orbifold cohomology\drp\ of the underlying
(algebraic) orbifold. The proof is based on a careful study of localization at
fixed points and of the resulting Koszul complexes. We provide examples of
Azumaya algebras for which this identification is, however, no longer valid. As
an example, we discuss some affine Weyl groups.Comment: Funding information adde
Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces
We define symmetric spaces in arbitrary dimension and over arbitrary
non-discrete topological fields \K, and we construct manifolds and symmetric
spaces associated to topological continuous quasi-inverse Jordan pairs and
-triple systems. This class of spaces, called smooth generalized projective
geometries, generalizes the well-known (finite or infinite-dimensional) bounded
symmetric domains as well as their ``compact-like'' duals. An interpretation of
such geometries as models of Quantum Mechanics is proposed, and particular
attention is paid to geometries that might be considered as "standard models"
-- they are associated to associative continuous inverse algebras and to Jordan
algebras of hermitian elements in such an algebra
Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra
While computer programs and logical theories begin by declaring the concepts
of interest, be it as data types or as predicates, network computation does not
allow such global declarations, and requires *concept mining* and *concept
analysis* to extract shared semantics for different network nodes. Powerful
semantic analysis systems have been the drivers of nearly all paradigm shifts
on the web. In categorical terms, most of them can be described as
bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style
completions from posets to suitably enriched categories. Yet it has been well
known for more than 40 years that ordinary categories themselves in general do
not permit such completions. Armed with this new semantical view of
Dedekind-MacNeille completions, and of matrix bicompletions, we take another
look at this ancient mystery. It turns out that simple categorical versions of
the *limit superior* and *limit inferior* operations characterize a general
notion of Dedekind-MacNeille completion, that seems to be appropriate for
ordinary categories, and boils down to the more familiar enriched versions when
the limits inferior and superior coincide. This explains away the apparent gap
among the completions of ordinary categories, and broadens the path towards
categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram
MacNeille completion and profinite completion can coincide on finitely generated modal algebras
Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by
piecing together results from various authors. Specifically, we show that if
is a residually finite, finitely generated modal algebra such that
has equationally definable principal
congruences, then the profinite completion of is isomorphic to its
MacNeille completion, and is smooth. Specific examples of such modal
algebras are the free -algebra and the free
-algebra.Comment: 5 page
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