 
Summary: Chapter 1
Categories and CPO's
(Notes for CS 819  February 15, 2000  c
fl2000 John C. Reynolds)
1.1 Relations
A relation is a set whose members are pairs. We will sometimes write
ae: x 7! x 0 or x ae x 0 as synonyms for [x; x 0 ] 2 ae, and describe this relation
ship by saying that ae relates x to x 0 .
The empty set is a relation, often called the empty relation. When two
relations satisfy ae ` ae 0 , ae is said to be a restriction of ae 0 , and ae 0 is said to be
an extension of ae.
When every pair [x; x 0 ] 2 ae satisfies x = x 0 , ae is said to be an identity
relation. For each set S there is an identity relation
I S
def
= f[x; x] j x 2 Sg;
called the identity relation on S. Moreover, every identity relation is I S for
a unique set S.
A variety of additional operations upon relations are significant. For
relations ae and ae 0 , we define
