Font and Jansana studied the Leibniz filters and the logic determined 
by the Leibniz filters of a given protoalgebraic
sentential logic. A filter is Leibniz when it is the smallest
among all the filters on the same algebra having the same
Leibniz congruence. Inspired by their work, a study
of the \(N\)-Leibniz theory systems of an \(N\)-protoalgebraic
\(\pi\)-institution is initiated. A theory system is \(N\)-Leibniz if it is
the smallest among all theory systems having the same Leibniz
\(N\)-congruence system. In this study,
some of the results of Font and Jansana on Leibniz filters are
adapted to cover the case of \(N\)-protoalgebraic \(\pi\)-institutions.
The \(N\)-Leibniz operator, used in the present setting,
is the operator associating with a
given theory family of a given \(\pi\)-institution the Leibniz
\(N\)-congruence system of the theory family, as introduced in previous
work by the author.