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.

Definition: Total Order and Chain

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.

Definitions: 1 2 3 4 5
Examples: 6
Explanations: 7
Lemmas: 8
Proofs: 9 10 11 12 13 14
Propositions: 15 16

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück