If the author wanted to describe a 'no ties' scenario where every object has its own unique place, they should have defined a strict total order.
They may know everything about mathematics for all I care. I am critiquing what I am reading, not the author's knowledge.
Edit: for anyone wanting a basic example, ["aa", "aa", "ab"] under the usual lexicographic <=. All elements are comparable, so "every object has its place depending on every other object." It also "leaves no room for ambiguity in terms of which element comes before which": aa = aa < ab. Linear order means everything is comparable, not that there are no ties. 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.
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.
"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.