Korean J. Math. Vol. 32 No. 2 (2024) pp.195-211
DOI: https://doi.org/10.11568/kjm.2024.32.2.195

Fuzzy lattice ordered group based on fuzzy partial ordering relation

Main Article Content

Sileshe Gone Korma
Radhakrishna Kishore Parimi
Dawit Chernet Kifetew

Abstract

In this paper, we introduce the concept of a fuzzy lattice ordered group, which is based on a fuzzy lattice that Chon developed in his paper "Fuzzy Partial Order Relations and Fuzzy Lattice". We will also discuss fuzzy lattice-ordered groups in detail, provide several results that are analogous to the classical theory of lattice-ordered groups, and characterize the relationship between a fuzzy lattice-ordered group using its level set and support. Moreover, we define the concepts of $fl$-subgroups, quotients, and cosets of $fl$-groups and obtain some fundamental results for these fuzzy algebraic structures.



Article Details

Supporting Agencies

Arba Minch University

References

[1] Mahmoud Bakhshi, On fuzzy convex lattice-ordered subgroups, Iranian Journal of Fuzzy Systems, 10 (3) (2013), 159–172. https://www.sid.ir/EN/VEWSSID/J_pdf/90820130310.pdf Google Scholar

[2] Garrett Birkhoff, Lattice theory, volume 25, American Mathematical Soc., 1940. http://dx.doi.org/10.2307/2268183 Google Scholar

[3] Garrett Birkhoff, Lattice-ordered groups, The Annals of Mathematics, 43 (2) (1942), 298–331. http://dx.doi.org/10.2307/1968871 Google Scholar

[4] Inheung Chon, Fuzzy partial order relations and fuzzy lattices, Korean Journal of Mathematics 17 (4) (2009), 361–374. https://www.dbpia.co.kr/Journal/articleDetail?nodeId=NODE08985027 Google Scholar

[5] M. R. Darnell, Theory of lattice-ordered groups, volume 1. Marcel Dekker, 1995. Google Scholar

[6] Parimi Radha, Krishna Kishore and Dawit Cherinet Kifetew, Properties of generalised lattice ordered groups, (IJCSAM) International Journal of Computing Science and Applied Mathematics 7 (1) (2021), 25–27. http://dx.doi.org/10.12962/j24775401.v7i1.7778 Google Scholar

[7] Valeri ̆ı Matveevich Kopytov and N Ya Medvedev, The theory of lattice-ordered groups, volume 307. Springer Science & Business Media, 2013. Google Scholar

[8] Sileshe Gone Korma, Radhakrishna Kishore Parimi, and Dawit Chernet Kifetew, Homomorphism and isomorphism theorems on fuzzy lattices, Research in Mathematics 10 (1) (2023), 2255411. http://dx.doi.org/10.1080/27684830.2023.2255411 Google Scholar

[9] Ivan Mezzomo, Benjamin C Bedregal, and Regivan HN Santiago, Types of fuzzy ideals in fuzzy lattices, Journal of Intelligent & Fuzzy Systems 28 (2) (2015), 929–945. http://dx.doi.org/10.3233/ifs-141374 Google Scholar

[10] Radha Krishna Kishore Parimi, Generalised lattice ordered groups (gl-groups), International Journal of Algebra 7 (2) (2013), 63–68. http://dx.doi.org/10.12988/ija.2013.13006 Google Scholar

[11] Ursala Paul and Paul Isaac, Fuzzy lattice ordered g-modules, International Journal of Fuzzy System Applications (IJFSA) 8 (3) (2019), 94–107. http://dx.doi.org/10.4018/ijfsa.2019070104 Google Scholar

[12] GSVS Saibaba, L-fuzzy prime spectrum of l-groups, Annals of Fuzzy Mathematics and Informatics 12 (2) (2016), 175–191. http://www.afmi.or.kr/articles_in_%20press/2016-02/AFMI-H-151116-2/AFMI-H-151116-2.pdf Google Scholar

[13] G.S.V.Satya Saibaba, Fuzzy lattice ordered groups, Southeast Asian Bulletin of Mathematics 32 (2008), 749–766. Google Scholar

[14] G.S.V.Satya Saibaba, Fuzzy convex sub l-groups, Annals of Fuzzy Mathematics and Informatics 11 (2016), 989–1001. http://www.afmi.or.kr/papers/2016/Vol-11_No-06/PDF/AFMI-11-6(989-1001)-H-151116-1R1.pdf Google Scholar

[15] Branimir Sˇeˇselja, Andreja Tepavˇcevi ́c, and Mirna Udoviˇci ́c, Fuzzy ordered structures and fuzzy lattice ordered groups, Journal of Intelligent & Fuzzy Systems 27 (3) (2014), 1119–1127. http://dx.doi.org/10.3233/ifs-131075 Google Scholar

[16] Stuart A Steinberg, Lattice-ordered rings and modules, volume 1. Springer, 2010. http://dx.doi.org/10.1007/978-1-4419-1721-8 Google Scholar

[17] J Vimala, Fuzzy lattice ordered group, International Journal of Scientific and Engineering Research 5 (9) (2014), 58–60. Google Scholar

[18] Lotfi A Zadeh, Fuzzy sets, Information and Control 8 (2) (1965), 338–353. http://dx.doi.org/10.1016/s0019-9958(65)90241-x Google Scholar

[19] Lotfi A Zadeh, Similarity relations and fuzzy orderings, Information sciences 3 (2) (1971), 177–200. http://dx.doi.org/10.1016/s0020-0255(71)80005-1 Google Scholar