  We study the isotonic regression estimator over a general countable
pre-ordered set. We obtain the limiting distribution of the estimator and study
its properties. It is proved that, under some general assumptions, the limiting
distribution of the isotonized estimator is given by the concatenation of the
separate isotonic regressions of the certain subvectors of an unrestrecred
estimator's asymptotic distribution. Also, we show that the isotonization
preserves the rate of convergence of the underlying estimator. We apply these
results to the problems of estimation of a bimonotone regression function and
estimation of a bimonotone probability mass function.
