Therefore, the only difference between a chain $(V,\preceq)$ and a poset $(V,\preceq)$ is that in a chain all elements are comparable, while in a poset some of them might be incomparable. This is the reason why the order "$\preceq$" is called "partial" in a poset, and "total" in a chain. Moreover, some subsets of a poset might be chains, but not vice versa. This leads us to the distinction between different cases when comparing the elements of a poset, which we want now to introduce:

Definition: Comparing the Elements of Posets and Chains

Let $(V,\preceq )$ be a poset or a chain. The comparison of any two given elements $a,b\in V$ can result in one of the following possibilities:

Definitions: 1
Lemmas: 2
Proofs: 3

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

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