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PETER H. KRAUSS AND DAVID M. CLARK
applications to algebra sheaves are only used as tools to construct a sub-
direct product, the structure of global sections of the sheaf. We shall de-
fine a very simple closure condition for subdirect products and call a subdi-
rect product global if it is closed in this respect. Subsequently we prove
the following:
Theorem The structures of global sections of sheaves are exactly the
global subdirect products of disjoint structures.
This characterization should completely dispense with those sheaf con-
structions in algebra whose only purpose is a global subdirect representation,
because it turns out that in these cases one always can verify directly that
some natural subdirect representation is global without rigging up the
cumbersome apparatus of a sheaf. This method by itself tends to simplify
matters considerably. Moreover it facilitates an analysis of the peculiar
"tightness" of global subdirect products which leads to the notion of
patching over a dual ring of subsets of the index set. This notion is at the
core of all sheaf representation. The rest is universal algebra.
Of course there are several well-known attempts in the literature to
analyse the sheaf construction in the setting of universal algebra (see, e.g.,
Davey [13] , Comer [8],Wolf [k^] and Burris and Werner [5]). However, each of
these explications has only limited applications to (admittedly large classes
of) special sheaf constructions known from the literature. There are other
authors who substantially contribute to the universal algebra of the sheaf
construction (see,e.g. Kennison [26], Ledbetter [29] and Werner [UO]).
Weispfenning [38] analyzes the sheaf construction in a model theoretic setting
and obtains results which are apparently closely related to ours. However,
Weispfenning's work has been carried out simultaneously with ours and has only
come to our attention after the completion of this manuscript. In general
we make no attempt to give an historically complete account of the sheaf con-
struction in algebra, or even to take into consideration every contribution in
this area. Our main goal is to present a unifying approach to the sheaf con-
struction in the setting of universal algebra. This approach is based on a
few simple ideas which we have extracted from the literature. The main pur-
pose of this paper is to demonstrate the universality of these ideas. This
also has guided our selection of applications. We have inspected far more
examples in the literature than we explicitly discuss in this study, or even
than we mention in the references. We have found none to which our methods do
not apply. Out of these examples we have chosen a few for actual presentation
which are generally considered significant results in sheaf representation and
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