A mistake concerning the ultra textit{LI}-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an textit{LI}-ideal to be an ultra textit{LI}-ideal are given. Moreover, the notion of an textit{LI}
-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) textit{LI}-ideal and an textit{ILI}-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper textit{LI}-ideal and Boolean textit{LI}-ideal, (2) prime proper textit{LI}-ideal and textit{ILI}-ideal, (3) proper obstinate textit{LI}-ideal, (4) ultra textit{LI}-ideal.
