# Total Preorder Relation

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A Total Preorder Relation is a Preorder Relation that is also a Total Relation.

**AKA:**Total Preorder.**See:**Total Order Relation.

## References

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Strict_weak_ordering#Total_preorders
- Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder \lesssim satisfies the following properties:
- For all x, y, and z, if x \lesssim y and y \lesssim z then x \lesssim z (transitivity).
- For all x and y, x \lesssim y or y \lesssim x (totality).
- Hence: for all x, x \lesssim x (reflexivity).

- Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder \lesssim satisfies the following properties: