Definition. Total order [dm-0007]

Liam O'Connor

A total order is a partial order that also satisfies totality.

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Definition. Chain [dt-000V]

Liam O'Connor

A chain in a poset $X$ is a totally ordered subset of $X$ . That is, a subset $Y \subseteq X$ is a chain iff: $\forall x, y \in Y.\; x \sqsubseteq y \lor y \sqsubseteq x$

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Definition. Partial order [dm-0000]

Liam O'Connor

A binary relation $(\sqsubseteq ) \subseteq X \times X$ is a partial order if it is, for all $x, y, z \in X$ :

reflexive: $x \sqsubseteq x$ ,
transitive: $x \sqsubseteq y$ and $y \sqsubseteq z$ implies $x \sqsubseteq z$ ,
antisymmetric: $x \sqsubseteq y$ and $y \sqsubseteq x$ implies $x = y$ .

Definition. Totality [dm-0008]

Liam O'Connor

A binary relation $\mathcal {R} \subseteq X \times X$ is total iff, for all $x, y \in X$ , either $x\ \mathcal {R}\ y$ or $y\ \mathcal {R}\ x$ .