Abstract
The maximal subalgebras and their intersection of a Lie algebra or a Lie superalgebra were studied by Racine, Scheiderer, Elduque, Melikyan, et al. The purpose of the present paper is to continue the investigation in order to obtain deeper structure theorems for Lie superalgebras. We develop the Frattini theory for Lie superalgebras, generalize Barnes's results to Lie superalgebras, and obtain some necessary and sufficient conditions for solvable Lie superalgebras and nilpotent Lie superalgebras. Moreover, some necessary and sufficient conditions for ϕ-free Lie superalgebras and elementary Lie superalgebras are given.
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