Tense Operators on BL-algebras and Their Applications

Akbar Paad

doi:10.18778/0138-0680.2021.11

Authors

Akbar Paad University of Bojnord, Department of Mathematics https://orcid.org/0000-0003-0929-9830

DOI:

https://doi.org/10.18778/0138-0680.2021.11

Keywords:

(simple) tense BL-algebra, tense operators, tense filter, tense congruence

Abstract

In this paper, the notions of tense operators and tense filters in \(BL\)-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense \(BL\)-algebras, tense \(MV\)-algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense filters of a \(BL\)-algebra is complete sublattice of \(F(L)\) of all filters of \(BL\)-algebra \(L\). Also, maximal tense filters and simple tense \(BL\)-algebras and the relation between them are studied. Finally, the notions of tense congruence relations in tense \(BL\)-algebras and strict tense \(BL\)-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided.

References

[1] M. Botur, I. Chajda, R. Halaš, M. Kolařík, Tense operators on basic algebras, International Journal of Theoretical Physics, vol. 50 (2011), pp. 3737–3749, DOI: https://doi.org/10.1007/s10773-011-0748-4
Google Scholar DOI: https://doi.org/10.1007/s10773-011-0748-4

[2] J. P. Burgess, Basic Tense Logic, [in:] D. M. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic, Springer Netherlands, Dordrecht (2002), pp. 1–42, DOI: https://doi.org/10.1007/978-94-017-0462-5_1
Google Scholar DOI: https://doi.org/10.1007/978-94-017-0462-5_1

[3] I. Chajda, Algebraic axiomatization of tense intuitionistic logic, Open Mathematics, vol. 9(5) (2011), pp. 1185–1191, URL: http://eudml.org/doc/269762
Google Scholar DOI: https://doi.org/10.2478/s11533-011-0063-6

[4] I. Chajda, M. Kolařík, Dynamic effect algebras, Mathematica Slovaca, vol. 62(3) (2012), pp. 379–388, DOI: https://doi.org/10.2478/s12175-012-0015-z
Google Scholar DOI: https://doi.org/10.2478/s12175-012-0015-z

[5] D. Diaconescu, G. Georgescu, Tense Operators on MV-Algebras and Łukasiewicz-Moisil Algebras, Fundamenta Informaticae, vol. 81(4) (2007), pp. 379–408.
Google Scholar

[6] A. V. Figallo, G. Gallardo, G. Pelaitay, Tense operators on m-symmetric algebras, International Mathematical Forum, vol. 41 (2011), pp. 2007–2014.
Google Scholar

[7] A. V. Figallo, G. Pelaitay, Tense operators on SHn-algebras, Pioneer Journal Algebra Number Theory and its Applications, vol. 1 (2011), pp. 33–41.
Google Scholar

[8] P. Hájek, Metamathematics of fuzzy logic, Springer Netherlands, Dordrecht (1988), DOI: https://doi.org/10.1007/978-94-011-5300-3
Google Scholar DOI: https://doi.org/10.1007/978-94-011-5300-3

[9] T. Kowalski, Varieties of tense algebras, Reports on Mathematical Logic, vol. 32 (1998), pp. 53–95.
Google Scholar

[10] C. Lele, J. B. Nganou, MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, vol. 218 (2013), pp. 103–113, DOI: https://doi.org/10.1016/j.fss.2012.09.014
Google Scholar DOI: https://doi.org/10.1016/j.fss.2012.09.014

[11] A. D. Nola, G. Georgescu, A. Iorgulescu, Pseudo BL-algebras: Part I, Multiple-Valued Logic, vol. 8(5–6) (2000), pp. 673–714.
Google Scholar

[12] A. D. Nola, L. Leuştean, Compact representations of BL-algebras, Archive for Mathematical Logic, vol. 42 (2003), pp. 737–761, DOI: https://doi.org/10.1007/s00153-003-0178-y
Google Scholar DOI: https://doi.org/10.1007/s00153-003-0178-y