On the Lattice of Filters of Intuitionistic Linear Algebras

Subject Areas : Transactions on Fuzzy Sets and Systems

Tenkeu Yannick Lea ¹ , Cyrille Nganteu ²

1 - Department of Mathematics, University of Yaound´e 1, Yaound´e, Cameroon.
2 - Department of Mathematics, University of Yaound´e 1, Yaound´e, Cameroon.

Received: 2022-08-24 Accepted : 2022-12-30 Published : 2023-05-01

Keywords: IL-algebra‎, ‎Filter‎, ‎Prime filter‎, ‎Congruence‎, ‎Residuated lattice.‎,

Abstract :

In this paper, we investigate the filter theory of Intuitionistic Linear Algebra (IL-algebra, in short) with emphasis on the lattice of filters of IL-algebras and relationship between filters and congruences on IL-algebras. We characterize the filter generated by a subset and give some related properties. The prime filter for IL-algebras is characterized and the prime filter theorem for IL-algebra is established. We get that the lattice (F (L), ⊆) of filters of an IL-algebra L is algebraic, Brouwerian, pseudocomplemented and endowed with the structure of Heyting algebra. We prove that the lattice of congruences and that of filters of any IL-algebra are isomorphic.

References:

[1] K. Blount and T. Tsinakis, The Structure of Residuated Lattices, International Journal of Algebra and Computation, 13(2003), 437-461.
[2] S. Burris and H. P. Sankappanavar, A course in Universal Algebra, Springer-Verlag, New York, (1981).
[3] D. Busneag, D. Piciu and A. Jeflea, Archimedian Residuated Lattices, Analele Stiintifice Ale Universitatii” AL.I. CUZA” DIN IASI (S.N) Mathematica, 16(2010), doi: 10.2478/v10157-010-0017-5.
[4] M. K. Chakraborty and J. Sen, MV-algebras embedded in CL-algebras, International journal of Approximate Reasoning, 18(1998), 217-229, doi.org/10.1016/S0888-613X(98)00007-3.
[5] R. Cretan and A. Jeflea, On the lattice of congruence filters of a residuated lattice, Annals of University of Craiova, Math. Comp. SCi. Ser., 33 (2006), 174-188.
[6] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Studies in Logic and the foundations of mathematics, Elsevier, 151(2007).
[7] J. Y. Girard, Linear Logic, Theoretical Computer Science, 5(1987), 1-101, doi.org/10.1016/0304-3975(87)90045-4.
[8] G. Gratzer, Lattice theory: Foundation, Springer Basel AG, (2011).
[9] S. Islam and A. Sanyal, A study on ideal of Classical Linear algebra, GANITA, 70(2020), 121-128.
[10] S. Islam, A. Sanyal and J. Sen, Filter theory of IL-algebras, Jr Cal Math. Soc., 16(2020), 113-126.
[11] S. Islam, A. Sanyal and J. Sen, Fuzzy filters of IL-algebras, Soft Computing, (2022), doi:10.1007/s00500-022-06985-1.
[12] S. Islam, A. Sanyal and J. Sen, Topological IL-algebras, Soft Computing, 26(2022), 8335-8349, doi.org/10.1007/s00500-022-07258-7.
[13] H. Rasiowa, An algebraic approch to non-classical logics, North Holland, Amsterdam, (1974).
[14] A. S. Troelstra, Lectures on Linear logic, Center for the Study of Language and Information, Standford, 29(1992).