Fantastic and associative filters in pseudo quasi-ordered residuated systems

Authors

DOI:

https://doi.org/10.14393/BEJOM-v4-2023-67770

Keywords:

Quasi-ordered residuated system (QRS), Pseudo-QRS, Filters in pseudo-QRS, Fantastic filter in pseudo QRS, Associative filter in pseudo QRS

Abstract

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.

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Author Biography

Daniel A. Romano, IMVI Banja Luka, Bosnia and Herzegovina

Daniel A. Romano was born in Visegrad, Bosnia and Herzegovina, where he finished primary (Elementary and Middle) and High school. At the Department of Mathematics of the Faculty of Sciences, University of Sarajevo ended the undergraduate studies in mathematics and physics (1969 - 1973) and graduate studies in mathematics (1976-1978). Doctorate in mathematics defended at the Faculty of Mathematics, University of Belgrade in 1986.

In the period 1978th-2017th crossed the teaching and scientific vocation of senior assistant (1978) to full professor (1997). He was a founder of the Mathematical Society of the Republic of Srpska (1994). He is also a member of the Scientific Society of Mathematicians Banja Luka (president), the International Mathematical Virtual Institute (general manager), European Mathematical Society and the International Mathematical Union. He was a mentor to several doctoral dissertations in mathematics and a large number of master works in the field of research in mathematics education. 

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.

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Published

2023-09-13

How to Cite

ROMANO, D. A. Fantastic and associative filters in pseudo quasi-ordered residuated systems. BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS, Uberlândia, v. 4, p. 1–17, 2023. DOI: 10.14393/BEJOM-v4-2023-67770. Disponível em: https://seer.ufu.br/index.php/BEJOM/article/view/67770. Acesso em: 23 jul. 2024.

Issue

Vol. 4 (2023): Brazilian Electronic Journal of Mathematics

Section

Articles - Pure Mathematics

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