164 Chapter 14
flips between the same two orders by going from the partial order to the strict order and
back again. Because of this, there is a one-to-one correspondence between partial orders
and strict orders, and whenever we refer to a partial order , we are entitled also to refer to
the corresponding strict order <, and vice versa.
With partial preorders, in contrast, which are reflexive transitive relations that are not
necessarily antisymmetric, this interdefinability correspondence breaks down. This is be-
cause there are different preorders
1
and
2
giving rise to the same strict order <. In this
sense, with preorders, the reflexive relation has more information. In the subject areas
using preorders, therefore, one should take the reflexive preorder as more fundamental than
the corresponding strict order.
To aid communication, mathematicians usually use a -like symbol for partial orders
and a <-like symbol for the corresponding strict partial order. For example, one might see
symbols like
≤ ⊆
used for a partial order, with the corresponding strict partial order denoted with the symbols
< .
To be clear, in order theory we use these symbols essentially as variables, writing , for
example, to represent whichever particular order we have in mind in that context. This
may be a somewhat more abstract use of this kind of symbol than some students have
encountered, which is why I remark specifically on it. In order theory, the symbol , say,
or , , and , do not necessarily have any fixed, predetermined interpretation or meaning;
these symbols represent different specific orders in different contexts, whichever is useful
for the argument at hand, just as the symbol x might refer to different real numbers or
objects in different arguments.
14.2 Minimal versus least elements
Let us study the following distinction for elements in a partial order.
Definition 120.
1. An element a in a partial order (A, )isaminimal element if there is nothing strictly
below it, that is, if there is no element b with b < a.
2. The element a is a least element, in contrast, if it is below all the other elements, so
a b for all b A.
Are these concepts the same? Is a minimal element the same thing as a least element?
Interlude. . .