On Semitopological De Morgan Residuated Lattices

Transactions on Fuzzy Sets and Systems

مقاله 9، دوره 2، شماره 1، مرداد 2023، صفحه 133-146   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2023.1966671.1047

نویسنده

Liviu-Constantin Holdon^*

Die Fakultt fr Unternehmertum, Ingenieurwissenschaften und Geschftsfhrung Ingenieurwissenschaften und Management Polytechnische Universitt Bukarest, Splaiul Independentei st., RO-060042 Bucharest (6), Romania E-mail: [e-mail] International Theoretical High School of Informatics Bucharest, Romania. E-mail: [e-mail]

چکیده

The class of De Morgan residuated lattices was introduced by L‎. ‎C‎. ‎Holdon (Kybernetika 54(3):443-475‎, ‎2018)‎, ‎recently‎, ‎many mathematicians have studied the theory of ideals or filters in De Morgan residuated lattices and some of them investigated the‎ ‎properties of De Morgan residuated lattices endowed with a topology‎. ‎In this paper‎, ‎we introduce the notion of semitopological De Morgan residuated lattice‎, ‎we present some examples and‎ ‎by considering the notion of upsets‎, ‎for any element $a$ of a De Morgan residuated lattice $L,$ there is a topology $\tau_{a}$ on $L$‎ ‎and we show that $L$ endowed with the topology‎ ‎$\tau_{a}$ is semitopological with respect to $\vee‎, ‎\wedge$ and $\odot,$ and right topological with respect to $\rightarrow.$‎ ‎Moreover‎, ‎in the general case of residuated lattices we prove that $L$ endowed with the topology‎ ‎$\tau_{a}$ is semitopological with respect to $\odot$ and right topological with respect to $\rightarrow.$‎ ‎Finally‎, ‎we obtain some of the topological‎ ‎aspects of this structure such as $L$ endowed with the topology‎ ‎$\tau_{a}$ is a $\mathbf{T_0}$-space‎, ‎but it is not a $\mathbf{T_1}$-space or Hausdorff space‎.

کلیدواژه‌ها

Residuated lattice‎؛ ‎De Morgan laws‎؛ ‎De Morgan residuated lattice‎؛ ‎Filter‎؛ ‎Semitopological algebras‎؛ ‎Hausdorff space‎

مراجع

[1] R. A. Borzooei, G. R. Rezaei and N. Kouhestani, On (semi) topological BL-algebra, Iran. J. Math. Sci. Inform., 6(1) (2011), 59-77.
[2] R. A. Borzooei and N. Kouhestani, On (semi)topological residuated lattices, Ann. Univ. Craiova, Math. Comp. Sc. Series, 41(1) (2014), 1529.
[3] D. Busneag, D. Piciu and A. M. Dina, Ideals in residuated lattices, Carpathian J. Math., 37(1) (2021), 53-63.
[4] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices: an algebraic glimpse at substructural logics, Stud. Logic Found. Math., Elsevier, (2007).
[5] P. Hajek, Mathematics of Fuzzy Logic, Dordrecht: Kluwer Academic Publishers, (1998).
[6] L. C. Holdon, On ideals in De Morgan residuated lattices, Kybernetika, 54(3) (2018), 443-475.
[7] L. C. Holdon, New topology in residuated lattices, Open Math., 16 (2018), 1-24.
[8] L. C. Holdon, The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices, Open Math., 18 (2020), 1206-1226.
[9] L. C. Holdon and A. Borumand Saeid, Ideals of Residuated Lattices, Stud. Sci. Math. Hung., 58(2) (2021), 182-205.
[10] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continnuos t-norms, Fuzzy Sets Syst., 124(3) (2001), 271-288.
[11] A. Iorgulescu, Algebras of logic as BCK algebra, Romania: Academy of Economic Studies Bucharest, (2008).
[12] S. Jenei and F. Montagna, A proof of standard completeness for Esteva and Godos logic MTL, Stud. Log., 70 (2002), 183-192.
[13] T. Kowalski and H. Ono, Residuated lattices: An algebraic glimpse at logics without contraction, JAIST, (2002).
[14] J. R. Munkres, Topology: a  rst course, Prentice-Hall, (1974).
[15] D. Piciu, Algebras of Fuzzy Logic, Craiova: Editura Universitaria Craiova, (2007).
[16] D. Piciu, Prime, minimal prime and maximal ideals spaces in residuated lattices, Fuzzy Sets Syst., 405 (2021), 47-64.
[17] E. Turunen, Mathematics Behind Fuzzy logic, New York: Physica-Verlag, (1999).
[18] M. Ward and R. P. Dilworth, Residuated lattices, Trans. Am. Math. Soc., 45 (1939), 335-354.
[19] F. Woumfo, B. B. Koguep Njionou, R. T. A. Etienne and L. Celestin, On State Ideals and State Relative Annihilators in De Morgan State Residuated Lattices, Int. J. Math. Sci., (2022). Available online: https://doi.org/10.1155/2022/6213448
[20] O. Zahiri and R. A. Borzooei, Semitopological BL-algebras and MV-algebras, Demonstr. Math., 47(3) (2014), 522-537.