We introduce a generic definition of reduction orderings on term
algebras containing associative-commutative (AC) operators. These
orderings are *compatible* with the AC theory
hence makes them suitable for use in deduction systems where AC
operators are built-in. Furthermore, they have the nice property of
being *total* on AC classes of ground terms, a required property
for example to avoid failure in AC-completion, or to insure
completeness of ordered strategies in first-order theorem proving
with built-in AC operators. We show that the two definitions already
known of such total and AC-compatible
orderings [narendran91rta,nieuwenhuis93rta] are actually
instances of our definition. Finally, we find new such orderings
which have more properties, first an ordering based on an integer
polynomial interpretation, answering positively to a question left
open by Narendran and Rusinowitch, and second an ordering which
allow to orient the distributivity axiom in the usual way, answering
positively to a question of the RTA'93 open problems
list.