Full Idea
A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx).
Gist of Idea
A 'linear or total ordering' must be transitive and satisfy trichotomy
Source
Herbert B. Enderton (Elements of Set Theory [1977], 3:62)
Book Reference
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.62