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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