Extreme fuzzy ideals and its applications on De Morgan residuated lattices

Abstract

The variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices (see L.C. Holdon [7]). X. Zhu, J. Yang and A. Borumand Saeid [16] used a special family of extreme fuzzy filters $\mathcal{F}$ on a BL-algebra L, they constructed a uniform structure $(L,\mathcal{K}),$ and then the part $\mathcal{K}$ induced a uniform topology ${\tau }_{\mathcal{F}}$ in L . Also, they proved that the pair $(L,{\tau }_{\mathcal{F}})$ is a topological BL-algebra, and some properties of $(L,{\tau }_{\mathcal{F}})$ were investigated. Inspired by their study, in this paper, we define the family of extreme fuzzy ideals $\mathcal{I}$ on a De Morgan residuated lattice L, we construct a uniform structure (L, K) , and then the part K induce a uniform topology ${\tau }_{\mathcal{I}}$ in L . We prove that the pair $(L,{\tau }_{\mathcal{I}})$ is a Topological De Morgan Residuated lattice, and some properties of $(L,{\tau }_{\mathcal{I}})$ are investigated. In particular, we show that $(L,{\tau }_{\mathcal{I}})$ is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, we give some characterizations of topological properties of $(L,{\tau }_{\mathcal{I}}).$ We note that, since ideals and filters are dual in BL-algebras (see C. Lele and J. B. Nganou [12]), a study on extreme fuzzy ideals in BL-algebras follows by duality, but in the framework of De Morgan residuated lattices, which is a larger class than BL-algebras, the duality between ideals and filters does not hold, so the study of extreme fuzzy ideals in De Morgan residuated lattices becomes interesting from algebraic and topological point of view, and the results of X. Zhu, J. Yang and A. Borumand Saeid [16] become particular cases of our theory.