A definition of maximal element of set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of partially-ordered set.
Target Context
- The reader will have a definition of maximal element of set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any partially-ordered set, \(\langle S, R \rangle\), any element, \(p \in S\), such that there is no element, \(p' \in S\), such that \(p R p'\)
2: Note
There may be multiple maximal elements when \(R\) is properly partial (meaning non-linear), because 2 maximal elements, \(p_1, p_2 \in S\), may be just not related.
For any linearly-ordered set, which is a kind of partially-ordered set, there can be no multiple maximal elements, because if \(p_1, p_2 \in S\) were maximal, exclusively \(p_1 R p_2\), \(p_1 = p_2\), or \(p_2 R p_1\), but the 1st and the 3rd cases are impossible, because \(p_1\) or \(p_2\) respectively would not be maximal.
