Stable Topology on Ideals for Residuated Lattices

Subject Areas : Transactions on Fuzzy Sets and Systems

Ariane GABRIEL Tallee Kakeu ¹ , Luc E. Diekouam ² , Blaise Bleriot Koguep N. ³ , Celestin Lele ⁴ , Daniel Akume ⁵

1 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
2 - Department of Mathematics, University of Maroua, HTTC Maroua, Maroua, Cameroon.
3 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
4 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
5 - Department of Computer Science, University of Buea, HTTTC Kumba, Kumba, Cameroon.

Received: 2024-01-14 Accepted : 2024-05-02 Published : 2024-05-07

Keywords: Residuated Lattices,

Abstract :

Residuated lattices are the major algebraic counterpart of logics without contraction rule, as they are more generalized logic systems including important classes of algebras such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and De Morgan residuated lattices among others, on which filters and ideals are sets of provable formulas. This paper presents a meaningful exploration of the topological properties of prime ideals of residuated lattices. Our primary objective is to endow the set of prime ideals with the stable topology, a topological framework that proves to be more refined than the well-known Zariski topology. To achieve this, we introduce and investigate the concept of pure ideals in the general framework of residuated lattices. These pure ideals are intimately connected to the notion of annihilator in residuated lattices, representing precisely the pure elements of quantales. In addition, we establish a relation between pure ideals and pure filters within a residuated lattice, even though these concepts are not dual notions. Furthermore, thanks to the concept of pure ideals, we provide a rigorous description of the open sets within the stable topology. We introduce the i-local residuated lattices along with their properties, demonstrating that they coincide with local residuated lattices. The findings presented in this study represent an extension beyond previous work conducted in the framework of lattices, and classes of residuated lattices.

References:

[1] Belluce LP. Semisimple algebras of infinite valued logic and bold fuzzy set theory. Canadian Journal of Mathematics. 1986; 38(6): 1356-1379. DOI: https://doi.org/10.4153/CJM-1986-069-0
[2] Holdon LC. On ideals in De Morgan residuated lattices. Kybernetika. 2018; 54(3): 443-475. DOI: https://doi.org/10.14736/kyb-2018-3-0443
[3] Kadji A, Lele C, Nganou JB, Tonga M. Folding theory applied to residuated lattices. International Journal of Mathematics and Mathematical Sciences. 2014; 2014: 1-12. DOI: https://doi.org/10.1155/2014/428940
[4] Ward M, Dilworth RP. Residuated lattices. Proceedings of the National Academy of Sciences. 1938; 24(3): 162-164. https://doi.org/10.1073/pnas.24.3.162
[5] Pavelka J. On fuzzy logic II. Enriched residuated lattices and semantics of propositional calculi. Mathematical Logic Quarterly. 1979; 25(2):119-134. DOI: https://doi.org/10.1002/malq.19790250706
[6] Zhu Y, Xu Y. On filter theory of residuated lattices. Information Sciences. 2010; 180(19): 3614-3632. https://doi.org/10.1016/j.ins.2010.05.034
[7] Gasse BV, Deschrijver G, Cornelis C, Kerre EE. Filters of residuated lattices and triangle algebras. Information Sciences. 2010; 180(16): 3006-3020. DOI: https://doi.org/10.1016/j.ins.2010.04.010
[8] Rachunek J, Salounova D. Ideals and involutive filters in generalizations of fuzzy structures. Fuzzy Sets and Systems. 2017; 311: 70-85. DOI: https://doi.org/10.1016/j.fss.2016.03.004
[9] Busneag C, Piciu D. The stable topology for residuated lattices. Soft Computing. 2012; 16: 1639-1655. DOI: https://doi.org/10.1007/s00500-012-0849-x
[10] Busneag D, Piciu D, Holdon L. Some properties of ideals in Stonean residuated lattices. Journal of Multi-valued Logic and Soft Computing. 2015; 24(5-6): 529-546.
[11] Luo QJ. Ideals in residuated lattices. Advances in Intelligent Systems and computing. 2016; 510: 407-414. DOI: https://doi.org/10.1007/978-3-319-46206-6 39
[12] Liu Y, Qin Y, Qin X, Xu Y. Ideals and fuzzy ideals on residuated lattices. International Journal of Machine Learning and Cybernetics. 2017; 8: 239-253. DOI: https://doi.org/10.1007/s13042-014-0317-2
[13] Piciu D. Prime, minimal prime and maximal ideals spaces in residuated lattices. Fuzzy Sets and Systems. 2021; 405: 47-64. DOI: https://doi.org/10.1016/j.fss.2020.04.009
[14] Mihaela I. Pure ideals in residuated lattices. Transactions on Fuzzy Sets and Systems. 2022; 1(1): 42-58. DOI: https://doi.org/10.30495/tfss.2022.690290
[15] Cavaccini V, Cella C, Georgescu G. Pure ideals of MV-algebras. Math. japonica. 1997; 45(2): 303-310. DOI: https://doi.org/10.2989/16073600009485976
[16] Belluce LP, Sessa S. The stable topology for MV-algebras. Quaestiones Mathematicae. 2000; 23(3): 269-277. DOI: https://doi.org/10.2989/16073600009485976
[17] Busneag D, Piciu D, Dina AM. Ideals in residuated lattices. Carpathian Journal of mathematics. 2021; 37(1): 53-63. DOI: https://doi.org/10.37193/CJM.2021.01.06
[18] Georgescu G. Reticulation of a quantale, pure elements and new transfer properties. Fuzzy Sets and Systems. 2022; 442: 196-221. DOI: https://doi.org/10.1016/j.fss.2021.06.005
[19] Rosenthal KI. Quantales and Their Applications. USA: Longman Scientific and Technical; 1990.
[20] Blyth TS. Lattices an Ordered Algebraic Structures. London: Springer; 2005.
[21] Johnstone PT. Stone Spaces. Cambridge university press; 1982.
[22] Georgescu G, Voiculescu I. Some abstract maximal ideal-like spaces. Algebra Universalis. 1989; 26: 90- 102. DOI: https://doi.org/10.1007/BF01243875
[23] Holdon LC. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices. Open Mathematics. 2020; 18: 1206-1226. DOI: https://doi.org/10.1515/math-2020-0061
[24] Tallee Kakeu AG, Koguep Njionou BB, Lele C, Strungmann ¨ L. Relative annihilators in bounded commutative residuated lattices. Indian Journal of Pure and Applied Mathematics. 2023; 54: 359-374. DOI:
https://doi.org/10.1007/s13226-022-00258-1
[25] Tallee Kakeu AG, Str¨ungmann L, Koguep Njionou BB, Lele C. α-ideals in bounded commutative residuated lattices. New Mathematics and Natural computation. 2023; 19(03): 611-630. DOI: https://doi.org/10.1142/S1793005723500254
[26] DE Marco G. Projectivity of pure ideals. Rendiconti del Seminario Matematico della Universit di Padova. 1983; 69: 289- 304. DOI: https://doi.org/10.1016/j.ins.2010.04.010
[27] Georgescu G, Voiculescu I. Isomorphic sheaf representations of normal lattices. Journal of Pure and Applied Algebra. 1987; 45(3): 213-223. DOI: https://doi.org/10.1016/0022-4049(87)90071-5
[28] Lele C, Nganou JB. MV-algebras derived from ideals in BL-algebras. Fuzzy Sets and Systems. 2013; 218: 103-113. DOI: https://doi.org/10.1016/j.fss.2012.09.014
[29] Yinga FT, Koguep Njionou BB, Lele C, Temgoua Alomo E. n-fold boolean, implicative and integral ideals on bounded commutative residuated lattices. New Mathematics and Natural Computation. 2019; 15(03): 427-445. DOI: https://doi.org/10.1142/S1793005719500248
[30] Cattaneo G, Ciucci D. Lattices with interior and closure operators and abstract approximation spaces. Transactions on Rough Sets X. 2009; 67-116. DOI: https://doi.org/10.1007/978-3-642-03281-3-3
[31] Rasoulli S, Dehgani A. The pure spectrum of a residuated lattice. Fuzzy Sets and Systems. 2023; 469: 108836. DOI: https://doi.org/10.1016/j.fss.2023.108636
[32] Yinga FT, Koguep Njionou BB, Temgoua Alomo E. n-fold fantastic and n-fold involutive ideals in bounded commutative residuated lattices. Discussiones Mathematicae General Algebra and Applications. 2022; 42(2): 363-381. DOI: https://doi.org/10.7151/dmgaa.1396