  In this paper we consider the logics $L_n^i$ obtained from the (n+1)-valued
Lukasiewicz logics $L_{n+1}$ by taking the order filter generated by i/n as the
set of designated elements. In particular, the conditions of maximality and
strong maximality among them are analysed. We present a very general theorem
which provides sufficient conditions for maximality between logics. As a
consequence of this theorem it is shown that $L_n^i$ is maximal w.r.t. CPL
whenever n is prime. Concerning strong maximality between the logics $L_n^i$
(that is, maximality w.r.t. rules instead of axioms), we provide algebraic
arguments in order to show that the logics $L_n^i$ are not strongly maximal
w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one
extension between $L_n^i$ and CPL obtained by adding to $L_n^i$ a kind of
graded explosion rule. Finally, using these results, we show that the logics
$L_n^i$ with n prime and i/n < 1/2 are ideal paraconsistent logics.
