Let be a poset and . The join of is the supremum of , if it exists. It is denoted by
if the elements of are indexed by a set :
In other words, , where the equality is directed in the sense that one side is defined iff the other side is, and when this is the case, both sides are equal. Dually, one defines the meet of to be the infimum of , if it exists. The meet of is denoted by .
Remark. The concepts of and of an ordered set are identical. Besides being notationally distinct, is often used in order theory, while is more prevalent in analysis. Moreover, is generally being viewed as a (partial) function on the powerset of the poset , while is frequently seen as an operation on sequences (or more generally nets) of elements of .
If , then , provided that both joins exist. We also have a dual statement: implies that , provided that both meets exist.
exists iff has a bottom , and when this is the case, . This is essentially the result of the previous bulleted statement. Dually, has a top iff exists, and when this is the case .
Simiarly and , where the equality is directed on both sides.
Let be a poset such that is defined for all subsets (of ) of cardinality , is it true that is defined for all subsets of cardinality ? The answer is no, even when is finite. A counterexample can be constructed as follows.
Let be an infinite chain with a top element (this can be found by taking the set of natural numbers and dualize the usual order). Adjoin three elements to so that and are below all elements of , and is covered by , and no two of are comparable. This new poset has the property that any three distinct elements have a join. For example, . However, does not exist.
|Date of creation||2013-03-22 17:27:53|
|Last modified on||2013-03-22 17:27:53|
|Last modified by||CWoo (3771)|