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 $y\ \mathcal {R}\ x$ .

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Theorem. Chain → Directed [dt-0012]

Liam O'Connor

All non-empty chains are directed.

Proof.

Liam O'Connor

Let $C$ be a chain. Then, given two elements $x,y \in C$ we know from totality that either $x \sqsubseteq y$ or $y \sqsubseteq x$ . If $x \sqsubseteq y$ , then $y$ is the upper bound, otherwise $x$ is the upper bound.

Definition. Total order [dm-0007]

Liam O'Connor

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