Volume 26 • Issue 2 • PP: 164-181 • 2025
Principal L-fuzzy ideals and filters on a trellis
View PDF Abstract
In this paper, we study the notion of principal (crisp) fuzzy ideals (resp. filters) on the setting of trellises (or weakly associative lattices as called by several authors). More specifically, we introduce the notions of L-fuzzy ideals and L-fuzzy filters on a given trellis and provide basic characterizations of these notions based on their weakly associative meet and join operations. We pay particular attention to the kind of principal L-fuzzy ideals (resp. filters) on a given trellis, which are more complicated in the absence of the (associativity) transitivity property.
Keywords
References
[1] S.P. Bhatta, H. Shashirekha, Some characterizations of completeness for trellises in terms of joins of cycles, Czechoslovak Mathematical Journal, vol. 54, pp. 267–272, 2004.
[2] H. A. Abu-Donia and S. S. Hassan, “Fixed points in complete fuzzy metric spaces using common property (E.A.),” Mathematics, vol. 9, no. 8, p. 867, 2021.
[3] S.P. Bhatta, S. George, Some fixed point theorems for pseudo ordered sets, Algebra and Discrete Mathematics, vol. 11, no. 1, pp. 17–22, 2011.
[4] Y. Bo, W. Wangming, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems, vol. 35, no. 2, pp. 231–240, 1990.
[5] S. Boudaoud, L. Zedam, S. Milles, Principal intuitionistic fuzzy ideals and filters on a lattice, Discussiones Mathematicae General Algebra and Applications, vol. 40, no. 1, pp. 75–88, 2020.
[6] N. Bourbaki, Topologie g´en´erale, Springer-Verlag, Berlin Heidelberg, 2007.
[7] B.A. Davey, H.A. Priestley, Introduction to trelliss and Order. Second ed, Cambridge University Press, Cambridge, 2002.
[8] B. Davvaz, O. Kazanci, A new kind of fuzzy Sublattice (Ideal, Filter) of a lattice, Int. J. Fuzzy. Syst., vol. 13, no. 1, pp. 55–63,2011.
[9] T. Yang, L. Liu, and C. Xu, “On fuzzy closure operators in lattice theory,” Symmetry, vol. 12, no. 11, p. 1875, 2020.
[10] E. Fried, Tournaments and non-associative lattices, Ann. Univ. Sci. Budapest, Sect. Math, vol. 13, pp. 151–164, 1970.
[11] E. Fried, G. Gr¨atzer, Some examples of weakly associative lattices, Colloquium Mathematicum, vol. 27, no. 2, pp. 215–221, 1973.
[12] K. Gladstien, Characterization of completeness for trellises of finite length, Algebra Universalis, vol. 3, pp. 341–344, 1973.
[13] J. A. Goguen, L-Fuzzy Sets, Journal of Mathematical Analysis and Applications, vol. 18, no. 1, pp. 145–174, 1967.
[14] G. Gr¨atzer, General Lattice Theory, Academic Press, New York, 1978.
[15] Y. Liu, K. Qin, Y. Xu, Fuzzy prime filters of lattice implication algebras, Fuzzy Information and Engineering, vol. 3, no. 3, pp. 235–246, 2011.
[16] I. Mezzomo, B.C. Bedregal and R.H.N. Santiago, Types of fuzzy ideals in fuzzy lattices, Journal of Intelligent and Fuzzy Systems, vol. 28, no. 2, pp. 929–945, 2015.
[17] S. Milles, L. Zedam, E. Rak, Characterizations of intuitionistic fuzzy ideals and filters based on lattice operations, Journal of Fuzzy Set Valued Analysis, vol. 2, pp. 143–159, 2017.
[18] H. Prodinger, Congruences defined by languages and filters, Information and Control, vol. 44, no. 1, pp. 36–46, 1980.
[19] B.S. Schr¨oder, Ordered Sets, Birkhauser, Boston, 2002.
[20] H. Skala, Trellis theory, Algebra Universalis, vol. 1, pp. 218–233, 1971.
[21] H. Skala, Trellis theory, American Mathematical Society, Providence, 1972.
[22] M. H. Stone, The theory of representation of Boolean algebras, Trans. Amer. Math. Soc., vol. 40, no. 1, pp. 37–111, 1936.
[23] M. A. Rodr´ıguez, J. R. Castro, and F. Moreno, “New approaches for fuzzy ideals and lattice structures in decision-making problems,” Applied Soft Computing, vol. 118, p. 108466, 2022.
[24] F. Vˇcelaˇr and Z. P´at´ıkov´a, On fuzzification of Tarski’s fixed point theorem without transitivity, Fuzzy Sets and Systems, vol. 320, pp. 93–113, 2017.
[25] B. Van Gasse, G. Deschrijver, C. Cornelis, E. Kerre, Filters of residuated lattices and triangle algebras, Inform. Sci., vol. 480, no. 16, pp. 3006–3020, 2010.
[26] S. Willard, General Topology, Addison-Wesley Publishing Company, Massachusetts, 1970.
[27] L.A. Zadeh, Fuzzy sets, Information and Control, vol. 8, pp. 331–352, 1965.
Cite This Article
Choose your preferred format