In this talk the logics Ln,i obtained from the (n+1)-valued Lukasiewicz logics by taking the order filter generated by i/n as the set of designated elements, will be discussed. In particular, the relationship of maximality and strong maximality among them will be analysed. A very general theorem which states sufficient conditions for maximality between logics will be presented. As a corollary of this theorem it will be shown that Lp,i is maximal w.r.t. CPL whenever p is prime. Concerning strong maximality between the logics Ln,i (that is, maximality w.r.t. rules instead of axioms), algebraic arguments will be given in order to shown that Lp,i is not strongly maximal w.r.t. CPL, even for p prime. Indeed, there is just one extension-by-rules between Lp,i and CPL, obtained by adding to Lp,i a kind of graded explosion rule. It will be also shown that the logics Lp,i for p prime and i/p < 1/2 are ideal paraconsistent logics in the sense of Arieli, Avron and Zamanski. The case of p=3 will be discussed in detail, presenting a 4-valued paraconsistent logic called J4 (together with a sound and complete Hilbert calculus for it) which generalizes the well-known 3-valued paraconsistent logic J3 introduced by D'Ottaviano and da Costa.