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].