Lattices of (Generalized) Fuzzy Ideals in Double Boolean Algebras
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Fernand Kuiebove Pefireko
1
(Department of Mathematics, University of Yaound´e 1, Yaound´e, Cameroon.)
کلید واژه: Double Boolean algebras, Fuzzy ideals, Fuzzy primary ideal.,
چکیده مقاله :
This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa). According to Rudolf Wille, a double Boolean algebra D := (D, ⊓, ⊔, :, ⌟, ?, ⊤) is an algebra of type (2, 2, 1, 1, 0, 0), which satisfies a set of properties. This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts. We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures. We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra D⊔.
This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa). According to Rudolf Wille, a double Boolean algebra D := (D, ⊓, ⊔, :, ⌟, ?, ⊤) is an algebra of type (2, 2, 1, 1, 0, 0), which satisfies a set of properties. This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts. We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures. We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra D⊔.
[1] M. Attallah, Complete Fuzzy Prime Ideals of Distributives Lattices, the journal of fuzzy Mathematics, 8(1) (2000), 151-156.
[2] T. Awoke, M. Rout, L. Mohandi and S. Satapathy, Bitcoin Price Prediction and Analysis Using Deep Learning Models, Networks and Systems, (2020), 1-5. DOI: https://doi.org/10.1007/978-981-15-5397-4-63
[3] A. Azari, P. Papapetrou, S. Denic and G. Peters, Cellular Traffic Prediction and Classification: a comparative evaluation
of LSTM and ARIMA, arXiv, (2019), 1-8. DOI: https://doi.org/10.48550/arXiv.1906.00939
[4] I. Chon, Fuzzy Partial Order Relations and Fuzzy Lattices, Korean Journal Mathematics, 17(4) (2009), 361-374.
[5] A. Fotso Tatuéné, M. Tonga, Transfer of structure between a universal algebra and its fuzzy algebra, (2016).
[6] S. Islam, A. Sanyal and J. Sen, Fuzzy filters of IL-algebras, Foundation, algebraic, and analytical methods in soft computing, Springer, (26) (2022), 7017-7027. DOI: https://doi.org/10.1007/s00500-022-06985-1
[7] B. B. N. Koguep, C. Nkuimi and C. Lele, On Fuzzy Primes Ideals of Lattices, SAMSA Journal of Pure and Applied Mathematics, (SJPAM), (3) (2008), 1-11.
[8] L. Kwuida, Prime ideal theorem for double Boolean algebra, Discussiones Mathematicae. General algebra and Applications, (27) (2007), 263-275. DOI: 10.7151/dmgaa.1130.
[9] P. Howlader and M. Banerjee, Remarks on Prime Ideal and Representation Theorems for Double Boolean Algebras, Proceedings of the 15th International conference on Concept Lattices and their Applications, CLA (2020), 83-94.
[10] I. Mezzomo, B. C. Bredegal, R. H. N. Santiago and R. H. S. Reiser, Fuzzy ideals of product operators on bounded fuzzy lattices, Second workshop-school of theoretical computer sciences, (2014), 1-7.
[11] Y. Tenkeu Jeufack, E. Temgoua Alomo and L. Kwuida, Filters Ideals and Congruences on Double Boolean Algebras, International conference on Formal Concept analysis, (2021), 270-280. DOI: https://doi.org/10.1007/978-3-030-77867-5-18
[12] K. Zhu, J. Wang and Y. Yang, Lattices of (Generalized) Fuzzy Ideals in Residuated Lattices International Journal of Applied Mathematics, 50(3) (2020).
