Let be a poset and a subset of . The upper set of is defined to be the set
and is denoted by . In other words, is the set of all upper bounds of elements of .
if , .
So is a closure operator.
An upper set in is a subset such that its upper set is itself: . In other words, is closed with respect to in the sense that if and , then . An upper set is also said to be upper closed. For this reason, for any subset of , the is also called the upper closure of .
Dually, the lower set (or lower closure) of is the set of all lower bounds of elements of . The lower set of is denoted by . If the lower set of is itself, then is a called a lower set, or a lower closed set.
is not the same as the set of upper bounds of , commonly denoted by , which is defined as the set . Similarly, in general, where is the set of lower bounds of .
When , we write for and for . and .
|Date of creation||2013-03-22 15:49:50|
|Last modified on||2013-03-22 15:49:50|
|Last modified by||CWoo (3771)|