◀ ▲ ▶Branches / Set-theory / Definition: Comparing the Elements of Posets and Chains
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:
- $a$ is equal to $b$: $b\preceq a$ and $a\preceq b$ (in this case, we also write $a=b$),
- $a$ is smaller than or equal to $b$: $a\preceq b$,
- $a$ is greater than or equal to $b$: $b\preceq a$ (in this case, we also write $a\succeq b$),
- $a$ is smaller than $b$: $a\preceq b$ and $a\neq b$ (in this case, we also write $a\prec b$),
- $a$ is greater than $b$: $b\succeq a$ and $a\neq b$ (in this case we also write $a\succ b$),
- $a$ and $b$ are incomparable (non-comparable), if none of the above relations hold.
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Adapted from CC BY-SA 3.0 Sources:
- Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück