Fantastic and associative filters in pseudo quasi-ordered residuated systems
DOI:
https://doi.org/10.14393/BEJOM-v4-2023-67770Keywords:
Quasi-ordered residuated system (QRS), Pseudo-QRS, Filters in pseudo-QRS, Fantastic filter in pseudo QRS, Associative filter in pseudo QRSAbstract
An interesting generalization of hoop-algebras and commutative residuated lattices is the concept of quasi-ordered residuated systems (shortly QRS) introduced in 2018 by Bonzio and Chajda. Quasi-ordered residuated system is an integral commutative monoid with two internal binary operations interconnected by a residuation connection. This specificity is the reason for the complexity of this algebraic structure and the existence of a significant number of substructures in it, such as various types of filters. The notion of pseudo quasi-ordered residuated systems was introduced and developed in 2022 by this author, omitting the commutativity requirement in QRSs, discussing, additionally, filters in it. Concept of pseudo QRSs is a generalization of the notion of QRSs. In this report, as a continuation of previous research, in addition to the introduction of concepts of fantastic and associative filters in a pseudo quasi-ordered residuated system, their mutual connection between them is discussed, and some examples are presented.
Downloads
References
ALAVI, S. Z., BORZOOEI, R. A. and KOLOGANI, M. A. Filter theory of pseudo hoop-algebras, Italian Journal of Pure and Applied Mathematics, v. 37, p. 619–632, 2017.
BONZIO, S. and CHAJDA, I. Residuated relational systems, Asian-European Journal of Mathematics, v. 11, n. 2, ID: 1850024, 2018. doi.org/10.1142/S1793557118500249
BORZOOEI, R. A. and KOLOGANI, M. A. Results on hoops, Journal of Algebraic Hyperstructures and Logical Algebras, v. 1, n. 1, p. 61–77, 2020. DOI:10.29252/hatef.jahla.1.1.5
BOSNACH, B. Komplementäre halbgruppen. Axiomatik und arithmetik, Fundamenta Mathematicae, v. 64, p. 257–287, 1969.
BOSNACH, B. Komplementäre halbgruppen. Kongruenzen und Quotienten, Fundamenta Mathematicae, v. 69, p. 1–14, 1970.
DI NOLA, A., GEORGESCU, G. and IORGULESCU, A. Pseudo-BL algebras: Part I, Multiple-Valued Logic, v. 8, n. 5-6, p. 673–714, 2002.
GEORGESCU, G. and IORGULESCU, A. Pseudo-MV algebras, Multiple-Valued Logic, v. 6, n. 1-2, p. 95–135, 2001.
GEORGESCU, G., LEU ̧STEAN, L. and PREOTEASA. Pseudo-hoops, Journal of Multiple-Valued Logic and Soft Computing, v. 11, n. 1-2, p. 153–184, 2005.
HART, J. B., RAFTER, L. and TSINAKIS, C. The structure of commutative residuated lattices, International Journal of Algebra and Computation, v. 12, n. 04, p. 509–524, 2002.
JUPSEN, P. and TSINAKIS, C. A survey of residuated lattices. In: NARTINEZ, J. (Ed.). Ordered algebraic structures, Kluwe Acad. Publ., Dordrecht 2002. p.19–56.
KOWALSKI, T. and ONO, H. Residuated Lattices: An Algebraic Glimpse atLogic without Contraction, Japan Advanced Institute of Science and Technology: Ishikawa, Japan, 2001.
ONO, H. Substructural logics and residuated lattices - an introduction. 50 Years of Studia Logica, Trednds in Logic, Kluwer Academic Publischer, v. 21, p. 193–228, 2003.
ROMANO, D. A. Pseudo-UP algebras. An introduction, Bulletin of International Mathematical Virtual Institute, v. 10, n. 2, p. 349–355, 2020. DOI:10.7251/BIMVI2002349R.
ROMANO, D. A. On a generalization of KU-algerbas: Pseudo-KU algebras. Open Journal of Mathematical Sciences, v. 4, p. 200–210, 2020. DOI:10.30538/oms2020.0111
ROMANO, D. A. Filters in residuated relational system ordered under quasi-order, Bulletin of International Mathematical Virtual Institute, v. 10, n. 3, p. 529–534, 2020. DOI: 10.7251/ BIMVI2003529R.
ROMANO, D. A. Associated filters in quasi-ordered residuated systems, Contributions to Mathematics, v. 1 p. 22–26, 2020. DOI: 10.47443/cm.2020.0010.
ROMANO, D. A. Implicative filters in quasi-ordered residuated system, Proyecciones Journal of Mathematics, v. 40, n. 2, p. 417–424, 2021. http://dx.doi.org/10.22199/issn.0717-6279-2021-02-0025
Romano, D. A. Comparative filters in quasi-ordered residuated system, Bulletin of the International Mathematical Virtual Institute, v. 11, n. 1, p. 177–184, 2021.DOI: 10.7251/ BIMVI2101177R
ROMANO, D. A. Normal filter in quasi-ordered residuated systems, Quasigroups and Related Systems, v. 30, n. 2, p. 317–327, 2022.
ROMANO, D. A. Pseudo quasi-ordered residuated systems, Pan-American Journal of Mathematics, v. 1, ID: 1-12, pp 1–10, 2022. https://doi.org/10.28919/cpr-pajm/1-12
WARD, M. and DILWORTH, R. P. Residuated lattices, Transactions of the American Mathematical Society, v. 45, p. 335–354, 1939.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
The published articles are licensed under the CreativeCommons CCBY-NC/4.0 version. By submitting the material for publication, the authors automatically waive their copyright, agree to the editorial guidelines of the journal, and assume that the text has been properly reviewed. Simultaneous submission of articles to other journals is prohibited, as is the translation of articles published in the journal into another language without proper authorization.