  The well-quasi-ordering (i.e., a well-founded quasi-ordering such that all
antichains are finite) that defines well-structured transition systems (WSTS)
is shown not to be the weakest hypothesis that implies decidability of the
coverability problem. We show coverability decidable for monotone transition
systems that only require the absence of infinite antichains and call well
behaved transitions systems (WBTS) the new strict superclass of the class of
WSTS that arises. By contrast, we confirm that boundedness and termination are
undecidable for WBTS under the usual hypotheses, and show that stronger
monotonicity conditions can enforce decidability. Proofs are similar or even
identical to existing proofs but the surprising message is that a hypothesis
implicitely assumed minimal for twenty years in the theory of WSTS can
meaningfully be relaxed, allowing more orderings to be handled in an abstract
way.
