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. . .