Ordering
I don't really know what an ordering is.
It has something to do with order and disorder, but I'm not clear on exactly what the relation is.
It is also an interesting fact that the verb "to order" means both to give a command, and to endow with order; to make something fit a certain pattern. I doubt that this is a coincidence. Like, to order some books is to impose your will on them, by making them fit a certain pattern.
Ordering seems like maybe it has something to do with time? At least one type of ordering does, namely time-ordering. Like, I think that the concepts of "before" and "after" are irreducible primaries. We just perceive that some events happen before or after other events. That probably comes first, and then by analogy we can start talking about other things being before or after.
Examples
Library books are ordered according to the Dewey Decimal System.
A single-file line of people could be ordered by height, with the shortest person in the front, and the tallest person in the back.
Natural numbers have a default ordering (by construction), where n+1 comes after n.
The marks on a clock are ordered clockwise.
To learn math, one must proceed in a certain order: some things must be learned before other things.
The alphabet is ordered. Little kids learn to remember this ordering by singing the alphabet song.
In feudal Europe, there was an order of succession (who inherits the throne after whom) which was largely agreed upon. The king's eldest son inherited the throne after the king died. If the king's eldest son was also dead, then his eldest son would inherit the throne. If the king's eldest son was dead and had no sons, then the king's second-eldest son would inherit the throne. Etc etc.
Non-examples
The books on my bookshelf are not ordered: they are in random positions with respect to one another.
Most single-file lines of people are not ordered by height.
Examples of "order" (n) which are not examples of "ordering" [TODO move]
The law of causality states that things act in accordance with their nature. One formulation of this law is that the universe has order.
If a riot happens, the police are supposed to come in and restore order to the streets.
The Order of the Phoenix.
The traditional concept
A set S is said to have the structure of a partial order is some binary relation R among members of the set (), which satisfying some properties. [TODO I don't want to get into partial ordering B.S. how to explain]
