On Universal Fields of Fractions for Free Algebras

Embedding Ordered Valued Domains into Division Rings

“…Moreover, by a result due to Lewin [10], it is isomorphic to the division ring generated by k < X > in k((G x )). By results of Lichtman [11,12] the division ring D constructed from k < X > using Theorem 4 is also isomorphic to k ( <X > ) . Hence using results from Section 3 we obtain: An open problem from the theory of ordered semigroups asks whether every ordering of a free monoid extends to the corresponding free group, see Revesz [19].…”

On embedding left-ordered groups into division rings

“…Further rather interesting examples such as 1 .M / where M is the complement of a fiber-type arrangement, surface groups, pure braid groups of closed surfaces, and free groups of finite rank of certain varieties are discussed in [4,20,43,44]. More on embedding residually torsion-free nilpotent groups into division rings via ultraproducts can be found for instance in [19,31].…”

Free skew fields have many ∗-orderings

“…Let L S be the free Lie algebra on a set S, then its enveloping algebra U(L S ) is isomorphic to the free associative algebra C S on S (Theorem 0.5 in [17] or a remark after Theorem 2.6.6 in [2]) and the field D(L S ) is isomorphic to the free field ∆ S on S (Theorem 1 in [14]). Note that the group ring C[F S ] of the free group F S on S contains C S and is contained in ∆ S .…”