For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict total order, which can equivalently be defined in two ways:
- a < b if and only if a ≤ b and a ≠ b
- a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)
Properties:
- The relation is transitive: a < b and b < c implies a < c.
- The relation is trichotomous: exactly one of a < b, b < a and a = b is true.
- The relation is a strict weak order, where the associated equivalence is equality.
We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can equivalently be defined in two ways:
- a ≤ b if and only if a < b or a = b
- a ≤ b if and only if not b < a
Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.
We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.
Read more about this topic: Total Order
Famous quotes containing the words strict, total and/or order:
“History creates comprehensibility primarily by arranging facts meaningfully and only in a very limited sense by establishing strict causal connections.”
—Johan Huizinga (1872–1945)
“Of lower states, of acts of routine and sense, we can tell somewhat; but the masterpieces of God, the total growths and universal movements of the soul, he hideth; they are incalculable. I can know that truth is divine and helpful; but how it shall help me I can have no guess, for so to be is the sole inlet of so to know.”
—Ralph Waldo Emerson (1803–1882)
“If we help an educated man’s daughter to go to Cambridge are we not forcing her to think not about education but about war?—not how she can learn, but how she can fight in order that she might win the same advantages as her brothers?”
—Virginia Woolf (1882–1941)