Fuzzy roughness in Lie algebra by reference point

Abstract

In this paper, we introduce the notion of reference point, lower and upper approximation with respect to reference point by a Lie algebra. We are concerned with some important properties of them.

For a fuzzy Lie subalgebra μ of a Lie algebra L, t-level relation U (μ, t) : = {(x, y) ∈ L × L : μ (x - y) ≥ t, μ ([x, y]) ≥ t} on L, and t-level relation with respect to the reference point a, U _e  (μ, t, a) : = {(x, y) ∈ L × L : μ ([a, x - y]) ≥ t}, are equivalence relations on L, for every t ∈ [0, 1] and every a ∈ L. We study lower and upper approximation with respect to the equivalence relations U (μ, t) and U _e  (μ, t, a) on a Lie algebra L, for every t ∈ [0, 1] and every a ∈ L. Furthermore, we show that if μ is a fuzzy Lie subalgebra of a Lie algebra L, a ∈ L and t ∈ [0, 1] then $\overline{\mathit{Fix}}\left(U\right(\mu ,t\left)\right)$ , $\underset{\_}{\mathit{Fix}}\left(U\right(\mu ,t\left)\right)$ , $\overline{\mathit{Fix}}\left(U_e\right(\mu ,t,a\left)\right)$ and $\underset{\_}{\mathit{Fix}}\left(U_e\right(\mu ,t,a\left)\right)$ are distributive complete lattices with respect to inclusion, where $\underset{\_}{\mathit{Fix}}\left(\theta \right)$ ( $\overline{\mathit{Fix}}\left(\theta \right)$ ) stand for the set of fixed points of upper (lower) equivalence relation θ. Also we obtain some relationship between ideals of a Lie algebra L and rough ideals with respect to the equivalence relations U _e  (μ, t, a) on L.