The growth of varieties of Leibniz algebras with nilpotent commutator subalgebra

It is proved that the exponents of certain varieties of Leibniz algebras with nilpotent commutator subalgebras exist and are integer.     

Almost nilpotent varieties with non-integer exponents do exist

The existence of an almost nilpotent variety of linear algebras with noninteger exponent is proved. Examples of almost nilpotent varieties with integer exponents were only known so far.     

The subvariety lattice of the join of two semigroup varieties

The join A ∨ B of two semigroup varieties A and B is investigated. The latrine of subvarieties of A ∨ B is completely described, It is shown that this lattice is finite and non-modular and that all varieties in it are finitely based and finitely generated.     

Infinite periodic words and almost nilpotent varieties

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. The existence of almost nilpotent varieties of arbitrary integer exponential growth with a fractional exponent is proved as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.     

Exponents of varieties of lie algebras with a nilpotent commutator subalgebra

We prove that the growth exponent of any variety of Lie algebras with a nilpotent commutator subalgebra is integral.     

On the intersection of conjugate nilpotent subgroups in permutation groups

We prove that, with exception of finitely many explicitly indicated cases, for a nilpotent subgroup in a symmetric or an alternating group there always exists a subgroup conjugate to it such that the intersection of the subgroups is trivial.