LM-fuzzy bitopological spaces

Abstract

In this paper, we introduce a new class of topology called an $LM$-{\it fuzzy bitopological spaces}. Also we introduces the closure operator $\tau_{1}\tau_{2}-Cl(A,r),$ and interior operator $\tau_{1}\tau_{2}-Int(A, r)$ in $LM$-fuzzy bitopological spaces as an extension of them in mated fuzzy bitopological spaces and establishes the relationship between $\tau_{1}\tau_{2}-Cl(A, r)$ and the smooth closure $\tau_{1}\tau_{2}-cl(A, r)$. Furthermore, we introduce different concepts of closed sets and open sets in double fuzzy bitopological spaces are extended to $LM$-fuzzy bitopological spaces and the algebraic structures associated with the families of these sets are investigated. As a result, certain monoids (denoted by $\tau_{1}\tau_{2}-O_{r}$, $\tau_{1}\tau_{2}-C_{r}$, $\tau_{1}\tau_{2}-L_{r}$, $\tau_{1}\tau_{2}-RO_{r}$, $\tau_{1}\tau_{2}-RC_{r}$, $\tau_{1}\tau_{2}-SO_{r}$,$\tau_{1}\tau_{2}-SC_{r}$, $\tau_{1}\tau_{2}-bO_{r}$, $\tau_{1}\tau_{2}-bC_{r}$, $\tau_{1}\tau_{2}-GO_{r}$, $\tau_{1}\tau_{2}-GC_{r}$, $\tau_{1}\tau_{2}-RGO_{r}$, $\tau_{1}\tau_{2}-RGC_{r}$, $\tau_{1}\tau_{2}-GSO_{r}$, $\tau_{1}\tau_{2}-GSC_{r}$ $\tau_{1}\tau_{2}-GbO_{r}$ and $\tau_{1}\tau_{2}-GbC_{r}$) contained in $L^{X}$ are identified. Finally, the study associates a lattice of monoids to each element of $M$, which is associative, complemented but not modular.