Orderability of subspaces of well-orderable topological spaces

We will show that all subspaces of well-ordered spaces are orderable, and we will characterize the well-orderability of subspaces of well-ordered spaces.     

First order theory of cyclically ordered groups

By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group H a pair $(G,z)$ where G is a totally ordered group and z is an element in the center of G, generating a cofinal subgroup $〈z〉$ of G, and such that the cyclically ordered quotient group $G/〈z〉$ is isomorphic to H. We first establish that, in this correspondence, the first-order theory of the cyclically ordered group H is uniquely determined by the first-order theory of the pair $(G,z)$. Then we prove that the class of cyclically orderable groups is an elementary class and give an axiom system for it. Finally we show that, in contrast to the fact that all theories of totally ordered Abelian groups have the same universal part, there are uncountably many universal theories of Abelian cyclically ordered groups.     

On Orderable Poly Engel Groups

It is known that an orderable n-Engel group is nilpotent. We show that an orderable group that is an extension of an n-Engel group by an n-Engel group is nilpotent-by-nilpotent. However, a finitely generated orderable poly n-Engel group need not be solvable in general.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.