Closure operations on Intuitionistic Linear algebras
Abstract
In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored.
Keywords
IL-algebra; closure operation; extended filter; prime filter
References
D. Busneag, D. Piciu, and A. Jeflea, Archimedean residuated lattices, An. Științ. Univ. Al. I. Cuza Iași. Mat. (N.S.) 56 (2010), no. 2, 227–252. DOI: 10.2478/v10157-010-0017-5
M.K. Chakraborty and J. Sen, MV-algebras embedded in a CL-algebra, Internat. J. Approx. Reason. 18 (1998), no. 3–4, 217–229. DOI: 10.1016/S0888-613X(98)00007-3
B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Second Ed., Cambridge University Press, New York, 2002.
J. Elliott, Rings, Modules and Closure Operations, Springer Monographs in Mathematics, Springer, Cham, 2019. DOI: 10.1007/978-3-030-24401-9
N. Epstein, A guide to closure operations in commutative algebra, in: C. Francisco et al. (eds.), Progress in Commutative Algebra 2, Walter de Gruyter GmbH & Co. KG, Berlin, 2012, pp. 1–37.
N. Galatos, P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and the Foundations of Mathematics, 151, Elsevier B. V., Amsterdam, 2007.
J.-Y. Girard, Linear logic, Theoret. Comput. Sci. 50 (1987), no. 1, 101 pp. DOI: 10.1016/0304-3975(87)90045-4
O.A. Heubo and J.B. Nganou, Closure operations on MV-algebras, Fuzzy Sets and Systems 418 (2021), 139–152.
A. Higuchi, Lattices of closure operators, Discrete Math. 179 (1998), no. 1–3, 267–272.
S. Islam, A. Sanyal, and J. Sen, Filter theory of IL-algebras, J. Calcutta Math. Soc. 16 (2020), no. 2, 113–126.
S. Islam, A. Sanyal, and J. Sen, Fuzzy filters of IL-algebras, Soft Comput. 26 (2022), no. 15, 7017–7027. DOI: 10.1007/s00500-022-06985-1
S. Islam, A. Sanyal, and J. Sen, Topological IL-algebras, Soft Comput. 26 (2022), no. 17, 8335–8349. DOI: 10.1007/s00500-022-07258-7
Y.C. Kim, Initial L-fuzzy closure spaces, Fuzzy Sets and Systems 133 (2003), no. 3, 277–297.
J.M. Ko and Y.C. Kim, Closure operators on BL-algebras, Commun. Korean Math. Soc. 19 (2004), no. 2, 219–232.
M. Kondo, Characterization of extended filters in residuated lattices, Soft Comput. 18 (2014), no. 3, 427–432. DOI: 10.1007/s00500-013-1100-0
D.S. Macnad, Modal operators on Heyting algebras, Algebra Universalis 12 (1981), no. 1, 5–29.
D. Salounova and J. Rachunek, A lattice-theoretical approch to extensions of filters in algebras of substructural logic, J. Algebr. Hyperstruct. Log. Algebras 3 (2022), no. 1, 5–14. DOI: 10.52547/HATEF.JAHLA.3.1.2
D. Spirito, Closure operations and star operations in commutative rings, Master’s Thesis, Università degli Studi Roma Tre, 2012.
Y.L.J. Tenkeu and C.T. Nganteu, On the lattice of filters of Intuitionistic Linear algebras, Trans. Fuzzy Sets Syst. 2 (2023), no. 1, 72–91. DOI: 10.30495/tfss.2022.1966200.1046
A.S. Troelstra, Lectures on Linear Logic, CSLI Lecture Notes, 29, Stanford University, Center for the Study of Language and Information, Stanford, CA, 1992.
P.V. Venkatanarasimhan, Stone’s topology for pseudocomplemented and bicomplemented lattices, Trans. Amer. Math. Soc. 170 (1972), 57–70.
M. Ward, The closure operators of a lattice, Ann. of Math. (2) 43 (1942), no. 2, 191–196.
Department of Mathematics, Faculty of Sciences, University of Yaoundé 1 Cameroon
https://orcid.org/0000-0003-3907-1546
Department of Mathematics, Ecole Normale Supérieure de Yaoundé, University of Yaoundé 1 Cameroon
Department of Mathematical Sciences, Saginaw Valley State University United States
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
