GLOBAL SUBDIRECT PRODUCTS
5
which are characteristic of the ideas and methods involved in the sheaf con-
struction. Finally, we have totally ignored some alternate approaches to the
sheaf construction, notably the category and topos oriented approaches. This
sufficiently illustrates the limitations of this study.
In Section 1 we fix notation and terminology and review a few well-known
facts which will be needed later. In Section 2 we characterize subdirect
products which arise from sheaf constructions and treat some well-known alge-
braic constructions in this framework, such as direct sums and Boolean powers.
Subsequently we convert some basic notions connected with the sheaf construction
into a form which is more suitable to our setting. In Section 3 we investigate
the hull-kernel topology in the setting of universal algebra. It turns out
that many sheaf constructions involve in some form or other the hull-kernel
topology induced by a subdirect representation. Section k contains the main
results of this study. We give a uniform method of constructing global sub-
direct products from the patching property. In Section 5 we consider subdi-
rect products which come from Hausdorff sheaves over Boolean spaces. Such
subdirect products are called Boolean, and we describe the special role played
by the normal transform in this setting. In Section 6 we pose the problem of
global subdirect representation of varieties and discuss some partial solu-
tions to the problem. In particular, we give
Kennisonfs
representation of
function rings [26], Ledbetter*s representation of vector groups and
relative Stone algebras (written communication) and Bulman-Fleming and Werner*s
representation of discriminator varieties [k] . In Sections 7 and 8 we give
examples from ring theory where sheaf representation has been most successful.
We have selected a representative sample from the vast literature in this area,
including Hofmann's results on semi-prime rings with identity [20],Dauns and
Hofmann's results on biregular rings [11] and weakly biregular rings [12],
Hofmann's results on Baer rings [20], Kennison's results on global subdirect
representation by integral domains [26] and Keimel's results on lattice-ordered
rings [2h].
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