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 £ h(i) J. For each
t(x,
0 =
n(x,
w)
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