On locally defined formations of soluble Lie and Leibniz algebras

Abstract: It is well-known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and that the latter has many local definitions. Over a field of non-zero characteristic, a saturated formation of soluble Lie algebras has at most one local definition but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.