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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