We define an ultra $LI$-ideal of a lattice implication algebra and give equivalent conditions for an $LI$-ideal to be ultra. We show that every subset of a lattice implication algebra which has the finite additive property can be extended to an ultra $LI$-ideal.

A mistake concerning the ultra $LI$-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an $LI$-ideal to be an ultra $LI$-ideal are given. Moreover, the notion of an $LI$-ideal is extended to $MTL$-algebras, the notions of a (prime, ultra, obstinate, Boolean) $LI$-ideal and an $ILI$-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 $LI$-ideal and Boolean $LI$-ideal,...