Page 27 - Advanced Linear Algebra
P. 27
Preliminaries 11
Similarly, a minimal element is an element 7 with the property that
there is no smaller element in , that is,
7
7Á ¬ ~
)
3 Let Á 7 . Then " 7 is an upper bound for and if
" and "
The unique smallest upper bound for and , if it exists, is called the least
upper bound of and and is denoted by lub ¸ Á . ¹
)
4 Let Á 7 . Then M 7 is a lower bound for and if
M and M
The unique largest lower bound for and , if it exists, is called the
greatest lower bound of and and is denoted by glb ¸ Á . ¹
Let be a subset of a partially ordered set . We say that an element 7 " 7 is
:
an upper bound for if " for all : . Lower bounds are defined
:
similarly.
Note that in a partially ordered set, it is possible that not all elements are
comparable. In other words, it is possible to have %Á & 7 with the property
that %& and & % .
Definition A partially ordered set in which every pair of elements is
comparable is called a totally ordered set , or a linearly ordered set . Any
totally ordered subset of a partially ordered set is called a chain in .
7
7
Example 0.6
)
s
1 The set of real numbers, with the usual binary relation , is a partially
ordered set. It is also a totally ordered set. It has no maximal elements.
)
2 The set o ~ ¸ Á Á Ã ¹ of natural numbers, together with the binary
relation of divides, is a partially ordered set. It is customary to write
to indicate that divides . The subset of consisting of all powers of
:
o
is a totally ordered subset of o , that is, it is a chain in o . The set
7 ~ ¸ Á Á Á Á Á ¹ is a partially ordered set under . It has two maximal
elements, namely and . The subset 8 ~ ¸ Á Á Á Á ¹ is a partially
ordered set in which every element is both maximal and minimal!
)
:
3 Let be any set and let ² F : ³ be the power set of , that is, the set of all
:
subsets of . Then ² F : ³ , together with the subset relation , is a partially
:
ordered set.
Now we can state Zorn's lemma, which gives a condition under which a
partially ordered set has a maximal element.
