  A binary relation over a free monoid is synchronous if it can be recognized
by a synchronous automaton that reads its two tapes simultaneously. We consider
the case where the free monoid is generated by a single element (which makes it
isomorphic to the additive monoid of integers) and where the binary relation
recognized is a strict order. Our main results are: given such an automaton it
is possible to determine whether or not is has infinite chains or antichains;
we characterize the orders that are linear; given two linear synchronous orders
we show how to determine whether or not they are equivalent.
