  Several insertion operations are studied applied to languages accepted by
one-way and two-way deterministic reversal-bounded multicounter machines. These
operations are defined by the ideals obtained from relations such as the
prefix, infix, suffix, and outfix relations, as well as operations defined from
inverses of a type of deterministic transducer with reversal-bounded counters
attached. The question of whether the resulting languages can always be
accepted by deterministic machines with the same number (or larger number) of
input-turns (resp., counters, counter-reversals, etc.) is investigated.
