Affiliations: [a] Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran
| [b] Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
Correspondence:
[*]
Corresponding author. Ali Akbar Estaji, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. E-mail: [email protected].
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, Ue (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 Ue (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
Fix¯(U(μ,t))
,
Fix_(U(μ,t))
,
Fix¯(Ue(μ,t,a))
and
Fix_(Ue(μ,t,a))
are distributive complete lattices with respect to inclusion, where
Fix_(θ)
(
Fix¯(θ)
) 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 Ue (μ, t, a) on L.
Keywords: Rough set, fuzzy Lie algebra, reference point, Lie ideal, fixed point
