Definition: An order is a set of elements, together with a binary relation between the elements of the set, which obeys certain laws.

  the relationship between elements in an order is commonly denoted as ≤ in formulas, but it can also be represented with an arrow from first object to the second.

All of the binary relations between the elements of your example are:

"aa" ≤ "aa"

"ab" ≤ "ab"

"aa" ≤ "ab"

> By claiming "no ties are permitted" while defining the order as a reflexive, antisymmetric relation, the author is mixing a strict-order intuition into a non-strict-order definition.

There aren't any ties to permit or reject.

  we can formulate it the opposite way too and say that each object should not have the relationship to itself, in which case we would have a relation than resembles bigger than, as opposed to bigger or equal to and a slightly different type of order, sometimes called a strict order.