such that 
and 
then A is called a directed set. In this case it means that given
any two arbitrary elements there is always a larger element available; thus
we say that the partial order is an upward directed set.21 March 2003
Order Structures II
by George E. Hrabovsky, President, MAST
Where We Have Been
We have been continuing our focus on the foundations of mathematics. We have discussed the basic ideas of set theory and logic, proof techniques, quantifiers, and have begun to develop the number system. We have laid a basic groundwork of sets, logic, and basic algebra. Last column we began to talk about order structures on sets.
Where Will We Go in This Column
We will explore some more ideas of ordering in sets.
Directed Sets
Last
week we examined the idea of a partial order. Suppose we have a partial
order, A, and
such that
and
then A is called a directed set. In this case it means that given
any two arbitrary elements there is always a larger element available; thus
we say that the partial order is an upward directed set.
We can also define a
partial order where given any two sets there is always a smaller set (or one
that precedes the two elements). This is called a downward directed set.
A lattice is a directed
set in both directions. It is interesting to note that, generally speaking,
a directed set need not be a lattice.
Directed sets are often indexed sets.
Well-Ordered Sets
If we have a partial
order, A, and 
,
then we call A well-ordered.
It is also true that every well ordered set is also a chain. The converse is not true. The reason for this is that some chains have no least element in them.
Comparing Order Structures; Order-Isomorphism
Given a set, T,
it is reasonable to assume that there will be occasions to compare order structures
within it. Let us say we have two subsets, 
and 
with
the order structures 
and 
.
Further, let's say that there is a bijection 
such that 
then we say that A and B are order-isomorphic. We can also
say that 
is an order-isomorphism.
Suggested Practice Problems
1. Give three examples
of directed sets. In what direction are they directed?
2. Give three examples of well-ordered sets.
3. Establish at least one order-isomorphism
Converted by Mathematica (March 21, 2003)