2
depend only on their order types. Then the values of f at
tuples of any given order type from A are either all the same
number
£
p, or all > p. In any event, at least one of the
numbers {0,.
..,p} is omitted by f on A. QED
We can immediately ask what else can be put there other than
the Boolean inequation fA ≠ Z. Think of Z as the universal
set.
This is what BRT = Boolean relation theory is all about. In
these lecture notes, we only consider “baby” BRT, where we
have just one function and one set.
1. SOME BOOLEAN ALGEBRA.
The notion of Boolean algebra is a very robust concept with
several different definitions. We give a particularly elegant
definition.
By way of motivation, the clearest examples of Boolean
algebras are the Boolean fields of sets. These are structures
(W,
∅
,S, ,
«
,’), where W is a family of subsets of the set S
closed under pairwise union, pairwise intersection, and
complement relative to S. Here ’ is complementation relative
to S.
We let
2
be the structure ({0,1},0,1,+,•,-), where x+y =
min(1,x+y), x•y is multiplication, and -x = 1-x.
A Boolean algebra is a structure B = (B,0,1,+,
•
,-), where 0,1
B, +,•:B2
Æ
B, -:B
Æ
B, such that every equation that
holds universally in
2
also holds universally in B.