A total order on a set is a way of ordering its elements to say
that some elements precede others, with the understanding that
any two elements can be compared one way or the other. A toset
is a set equipped with a total order. The category of finite
nonempty totally ordered sets and order-preserving maps is
called , the simplex category. The category of all finite
totally ordered sets and order-preserving maps is called , the
augmented simplex category. A linear order is much like a total
order, except that it is based on an irreflexive relation .
Using excluded middle, one can move between linear orders and
total orders using negation; that is, the negation of a total
order is a linear order and vice versa. Actually one usually
swaps the order too, as follows: One often sees defined as but ;
this is equivalent, but doesn't show the duality explicitly.
Similarly, one often sees defined as or ; this is not even
equivalent constructively, although it is classically. In
classical mathematics, the distinction be