GLOBAL SUBDIRECT PRODUCTS

7

0 X x ) ] J = {i € i I 01. N cp[7r.(x)]}

and for x, y e flA. we write

E(x, y) = lx = yK , D(x, y) = IT x * y J.

The ternary discriminator operation and the normal transform are defined by

z i f x = y

x i f x i y

z i f x = y

w i f x + y

respectively, t and n canonically induce operations t and n on

nAi | i € I:

t"(x, y, z)(i) = t(x(i), y(i), z(i))

n(x, y, z, w)(i) = n(x(i), y(i), z(i), w(i))

Our first topic for review is the Wallman-compactification of a topolo-

gical space. This compactification is constructed from the ring of closed

sets. The details are well-known and the pertinent facts are easy to prove

(Wallman [36]). Let I be a non-empty set and let f2, be a family of sub-

sets of I. 1R, is called a ring of subsets of I if

(i) 0 €fc;

(ii) if F, G € 71 then FUG, F f l G G 36 .

A ring of subsets of I is called Boolean if

(iii) if F, G€ IfL then F - G G *R, ,

and a Boolean ring of subsets of I is called a field if

(iv) I G K.

We say that 'frg , covers I if U *& = I and we define

^ = {I - F j F € ft}.

Let ^ be a ring of subsets of I and let J and M be the sets of

prime filters and maximal filters on ^ respectively. Since ^ is a

distributive lattice, M 5EJ- For each i € I define

h(i) = {F G 1R, I i G F}.

Since 0 Gfc, h(i) ± fc . Thus for every i G I, h(i) = 0 or h(i) € J.

Moreover, fc covers I if and only if for every i £ I» h(i) € J. For each

t(x,

0 =

n(x,

w)