In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad(A) contained in Fix(A).

The central kernel K(G) of a group G is the subgroup consisting of all elements fixed by every central automorphism of G. It is proved here that if G is a finite-by- nilpotent group whose central kernel has finite index, the G is finite over the centre, and the elements of finite order of G form a finite subgroup; in particular G is finite, provided that it is periodic. Moreover, if G is a periodic finite-b-nilpotent group and G/K(G) is a Cernikov group, it turns out that G itself is a Cernikov group.

We should like to construct all the semigroups that have a left univocal factorization with factors to pair (A,B) of prescribed semigroups such that their intersections consists of only one element, that is right identity of A and left identity of B.

An associative ring R, not necessarily with an identity element, is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that if the adjoint group of a semilocal ring R is locally supersoluble, then R is locally Lie supersoluble and its Jacobson radical is contained in a locally Lie nilpotent ideal of finite index in R.