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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