The lattice of prefilters of an EQ-algebra. (English) Zbl 1393.03055

Summary: In this paper, the notion of a prefilter generated by a nonempty subset of an EQ-algebra is introduced and a characterization of it is obtained. It is proved that the set of all prefilters of an EQ-algebra is an algebraic lattice and it is a Brouwerian lattice for an \(\ell\)EQ-algebra. Furthermore, it is shown that the set of all principal prefilters of an \(\ell\)EQ-algebra is a sublattice of the lattice of prefilters. Then by defining an implication between two prefilters, it is determined that the lattice of prefilters is a Heyting algebra, for an \(\ell\)EQ-algebra. Finally, the EQ-algebras for which the lattice of prefilters is a Boolean algebra are given.


03G25 Other algebras related to logic
06F35 BCK-algebras, BCI-algebras
Full Text: DOI


[1] Balbes, R.; Dwinger, P., Distributive Lattices (1974), University of Missouri Press: University of Missouri Press Columbia, Missouri, xiii + 294 pp · Zbl 0321.06012
[2] Chang, C. C., Algebraic analysis of many valued logics, Trans. Am. Math. Soc., 88, 467-490 (1958) · Zbl 0084.00704
[3] Dyba, M.; El-Zekey, M.; Novák, V., Non-commutative first-order EQ-logics, Fuzzy Sets Syst., 292, 215-241 (2016) · Zbl 1383.03057
[4] Dyba, M.; Novák, V., EQ-logics: non-commutative fuzzy logics based on fuzzy equality, Fuzzy Sets Syst., 172, 13-32 (2011) · Zbl 1229.03027
[5] El-Zekey, M., Representable good EQ-algebras, Soft Comput., 14, 1011-1023 (2010) · Zbl 1201.03061
[6] El-Zekey, M.; Novák, V.; Mesiar, R., On good EQ-algebras, Fuzzy Sets Syst., 178, 1-23 (2011) · Zbl 1242.03089
[7] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets Syst., 124, 271-288 (2001) · Zbl 0994.03017
[8] Gratzer, G., Lattice Theory. First Concepts and Distributive Lattices, A Series of Books in Mathematics (1972), W.H. Freeman and Company: W.H. Freeman and Company San Francisco
[9] Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer: Kluwer Dordrecht · Zbl 0937.03030
[10] Liu, L. Z.; Li, K. T., \(R_0\)-algebras and weak dually residuated lattice ordered semigroups, Czechoslov. Math. J., 56, 339-348 (2006) · Zbl 1164.06324
[11] Liu, L.; Zhang, X., Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst., 26, 2087-2097 (2014) · Zbl 1305.03061
[12] Novák, V., On fuzzy type theory, Fuzzy Sets Syst., 149, 235-273 (2005) · Zbl 1068.03019
[13] Novák, V.; De Baets, B., EQ-algebra, Fuzzy Sets Syst., 160, 2956-2978 (2009) · Zbl 1184.03067
[14] Pei, D. W., On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets Syst., 138, 187-195 (2003) · Zbl 1031.03047
[15] Turunen, E., Mathematics Behind Fuzzy Logic (1999), Physica-Verlag: Physica-Verlag Heidelberg · Zbl 0940.03029
[16] Zadeh, L. A., Is there a need for fuzzy logic?, Inf. Sci., 178, 2751-2779 (2008) · Zbl 1148.68047
[17] Zhan, J. M.; Xu, Y., Some types of generalized fuzzy filters of BL-algebras, Comput. Math. Appl., 56, 1604-1616 (2008) · Zbl 1155.06302
[18] Zhou, H. J.; Zhao, B., Stone-like representation theorems and three-valued letters in \(R_0\)-algebras (nilpotent minimum algebras), Fuzzy Sets Syst., 162, 1-26 (2011) · Zbl 1213.03081
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