A definition of linearly-ordered set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of relation.
Target Context
- The reader will have a definition of linearly-ordered 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
Any set with any relation, \(\langle S, R \rangle\), such that 1) \(R\) is a trichotomy for any element, \(p_1 \in S\): for any element, \(p_2, \in S\), exclusively \(p_1 R p_2\), \(p_1 = p_2\), or \(p_2 R p_1\); 2) \(R\) is transitive: for any elements, \(p_1, p_2, p_3 \in S\), such that \(p_1 R p_2\) and \(p_2 R p_3\), \(p_1 R p_3\)
2: Note
Any linearly-ordered set is a kind of partially-ordered set.
