820 research outputs found
Sup-lattice 2-forms and quantales
A 2-form between two sup-lattices L and R is defined to be a sup-lattice
bimorphism L x R -> 2. Such 2-forms are equivalent to Galois connections, and
we study them and their relation to quantales, involutive quantales and
quantale modules. As examples we describe applications to C*-algebras.Comment: 30 pages. Contains more detailed background section and corrections
of several typos and mistake
A note on infinitely distributive inverse semigroups
We show that in any infinitely distributive inverse semigroup the existing
binary meets distribute over all the joins that exist.Comment: 3 page
Quantales of open groupoids
It is well known that inverse semigroups are closely related to \'etale
groupoids. In particular, it has recently been shown that there is a
(non-functorial) equivalence between localic \'etale groupoids, on one hand,
and complete and infinitely distributive inverse semigroups (abstract complete
pseudogroups), on the other. This correspondence is mediated by a class of
quantales, known as inverse quantal frames, that are obtained from the inverse
semigroups by a simple join completion that yields an equivalence of
categories. Hence, we can regard abstract complete pseudogroups as being
essentially ``the same'' as inverse quantal frames, and in this paper we
exploit this fact in order to find a suitable replacement for inverse
semigroups in the context of open groupoids that are not necessarily \'etale.
The interest of such a generalization lies in the importance and ubiquity of
open groupoids in areas such as operator algebras, differential geometry and
topos theory, and we achieve it by means of a class of quantales, called open
quantal frames, which generalize inverse quantal frames and whose properties we
study in detail. The resulting correspondence between quantales and open
groupoids is not a straightforward generalization of the previous results
concerning \'etale groupoids, and it depends heavily on the existence of
inverse semigroups of local bisections of the quantales involved.Comment: 55 page
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