**
Czechoslovak Mathematical Journal, Vol. 57, No. 2, pp. 591-605, 2007
**

#
Ultra $LI$-Ideals in lattice implication algebras

and $MTL$-algebras

##
Xiaohong Zhang, Keyun Qin, Wieslaw A. Dudek

* Xiaohong Zhang*, Department of Mathematics, Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang Province, P. R. China, e-mail: ` zxhonghz@263.net`; * Keyun Qin*, Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China, e-mail: ` keyunqin@263.net`; * Wieslaw A. Dudek*, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland, e-mail: ` dudek@im.pwr.wroc.pl`

**Abstract:** 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, (2) prime proper $LI$-ideal and $ILI$-ideal, (3) proper obstinate $LI$-ideal, (4) ultra $LI$-ideal.

**Keywords:** lattice implication algebra, $MTL$-algebra, (prime, ultra, obstinate, Boolean) $LI$-ideal, $ILI$-ideal

**Classification (MSC 2000):** 03G10, 06B10, 54E15

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