No abstract

Neumann's totally ordered power series fields and division rings may be tipped over to form archimedean lattice-ordered fields and division rings. This process is described and then generalised to produce non-archimedean lattice-ordered fields and division rings in wliich 1 > 0 and, as well, ones in which 1^0 .

A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.1991 Mathematics subject classification (Amer. Math. Soc): primary 06 F 25; secondary 06 F 15, 12 J 15, 13 J 05, 16 A 27, 16 A 86.

Let (P, ) be a (nearly) uniform ordered space. Let (P, ) be the uniform completion of (P, ) at . Several partial orders for P are introduced and discussed. One of these orders provides an adjoint to the functor which embeds the category of uniformly complete uniform ordered spaces in the category of uniform ordered spaces, both categories with uniformly continuous order-preserving functions. When P is a join-semilattice with uniformly continuous join, two of these orders coalesce to the unique partial order with respect to which P is a join-semilattice, P is a join-subsemilattice of P, and the join on P is continuous.