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Total order
Definition
| $X$ |
| $ \le \in \text{TotalOrd}(X) $ |
| $ x\in \mathrm{dom}(\le) $ |
| $ x \le y\ \lor\ y\le x $ |
| $ (x\le y) \land (y\le x) \implies (x=y) $ |
| $ (x \le y) \land (y \le z) \Leftrightarrow (x\le z) $ |
Here we use infix notation: $x\le y\ \equiv\ \le(x,y)$.
Discussion
The relation $\le$ is an order relation if it's in the intersection of all reflexive, all anti-symmetric and all transitive relation. Hence
The first axiom $ x \le y\ \lor\ y\le x $ is called totality and implies $ x \le x $. Therefore a linear order is a partial order, although the converse is not true in general.
Reference
Wikipedia: Order theory
Context
Subset of
