Ultra LI-ideals in lattice implication algebras and MTL-algebras

TL;DR

This paper corrects previous misconceptions about ultra LI-ideals in lattice implication algebras, extends the concept to MTL-algebras, and establishes equivalences among various types of LI-ideals.

Contribution

It introduces new conditions for ultra LI-ideals, extends LI-ideals to MTL-algebras, and proves key equivalences among different LI-ideal notions.

Findings

Corrected previous errors regarding ultra LI-ideals.

Extended LI-ideals to MTL-algebras with new definitions.

Proved equivalences among prime, Boolean, obstinate, and ultra LI-ideals.

Abstract

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.