In a poset, it is not required that all elements are comparable. However, there are sets, in which this is the case, for instance, the set of natural numbers $(\mathbb N,\le)$ with the usual order relation $”\le”$. We want now to define such types of order relations and sets formally and give them another name.
The partial order $”\preceq”$ of a poset $(V,\preceq )$ is called a total order (or linear order), if “$\preceq$” is connex, i.e. if all pairs of elements $(a,b)\in V\times V$ can be ordered by $”\preceq”.$
A poset $(V,\preceq )$ with a total order $”\preceq”$ is called a chain.
| | | | | created: 2016-09-06 22:00:25 | modified: 2019-02-03 09:21:13 | by: bookofproofs | references: 
 Brenner, Prof. Dr. rer. nat., Holger: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014