  Polynomial closure is a standard operator which is applied to a class of
regular languages. In the paper, we investigate three restrictions called left
(LPol), right (RPol) and mixed polynomial closure (MPol). The first two were
known while MPol is new. We look at two decision problems that are defined for
every class C. Membership takes a regular language as input and asks if it
belongs to C. Separation takes two regular languages as input and asks if there
exists a third language in C including the first one and disjoint from the
second. We prove that LPol, RPol and MPol preserve the decidability of
membership under mild hypotheses on the input class, and the decidability of
separation under much stronger hypotheses. We apply these results to natural
hierarchies.
  First, we look at several language theoretic hierarchies that are built by
applying LPol, RPol and MPol recursively to a single input class. We prove that
these hierarchies can actually be defined using almost exclusively MPol. We
also consider quantifier alternation hierarchies for two-variable first-order
logic and prove that one can climb them using MPol. The result is generic in
the sense that it holds for most standard choices of signatures. We use it to
prove that for most of these choices, membership is decidable for all levels in
the hierarchy. Finally, we prove that separation is decidable for the hierarchy
of two-variable first-order logic equipped with only the linear order.
