  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.
