  We consider the problem of partitioning effectively a given symmetric (and
irreflexive) rational relation R into two asymmetric rational relations. This
problem is motivated by a recent method of embedding an R-independent language
into one that is maximal R-independent, where the method requires to use an
asymmetric partition of R. We solve the problem when R is realized by a
zero-avoiding transducer (with some bound k): if the absolute value of the
input-output length discrepancy of a computation exceeds k then the length
discrepancy of the computation cannot become zero. This class of relations
properly contains all recognizable, all left synchronous, and all right
synchronous relations. We leave the asymmetric partition problem open when R is
not realized by a zero-avoiding transducer. We also show examples of total
wordorderings for which there is a relation R that cannot be partitioned into
two asymmetric rational relations such that one of them is decreasing with
respect to the given word-ordering.
