Wiktionary
n. (context set theory English) A relation that is reflexive, antisymmetric, and transitive (i.e., that is a partial order), and having the property that for any two elements of its set, one is related to the other.
Wikipedia
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is transitive, antisymmetric, and total (this relation is denoted here by infix ≤ ). A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.
If X is totally ordered under ≤ , then the following statements hold for all a, b and c in X:
If a ≤ b and b ≤ a then a = b ( antisymmetry); If a ≤ b and b ≤ c then a ≤ c ( transitivity); a ≤ b or b ≤ a ( totality).Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the property of "totality" means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear. Totality also implies reflexivity, i.e., a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (It requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension of that partial order.
Usage examples of "total order".
She relayed plans to Chev via Veil, even though they stood not five feet apart.
The computers were losing track - with so many ships packed into so small a volume mutual interference made it impossible to generate an accurate drive field count - but Plotting estimated there were already ninety-plus Bug SDs in the system, and Fifth Fleet's total order of battle was only a hundred and seventeen ships, twenty percent of them tugs or antimissile escorts without a single offensive weapon.
The computers were losing track -- with so many ships packed into so small a volume mutual interference made it impossible to generate an accurate drive field count -- but Plotting estimated there were already ninety-plus Bug SDs in the system, and Fifth Fleet's total order of battle was only a hundred and seventeen ships, twenty percent of them tugs or antimissile escorts without a single offensive weapon.