  In this paper, we consider classes of decision tables closed under removal of
attributes (columns) and changing of decisions attached to rows. For decision
tables from closed classes, we study lower bounds on the minimum cardinality of
reducts, which are minimal sets of attributes that allow us to recognize, for a
given row, the decision attached to it. We assume that the number of rows in
decision tables from the closed class is not bounded from above by a constant.
We divide the set of such closed classes into two families. In one family, only
standard lower bounds $\Omega (\log $ ${\rm cl}(T))$ on the minimum cardinality
of reducts for decision tables hold, where ${\rm cl}(T)$ is the number of
decision classes in the table $T$. In another family, these bounds can be
essentially tightened up to $\Omega ({\rm cl}(T)^{1/q})$ for some natural $q$.
