Constructing lattice-ordered fields and division rings

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 .

Subfields of Lattice-Ordered Fields That Mimic Maximal Totally Ordered Subfields

“…the positive cone of a subfield which satisfies condition (1) or (2) is contained in the set determined by condition (3). And then in section 4, we find several conditions, each of which is equivalent to this set being the positive cone of an osubfield of (L, +, ·, ).…”

“…Then, as observed in [3], (L, +, ⊗ u , ) is an -field, where ⊗ u is defined by letting x ⊗ u y = xyu −1 . It is easy to see that the multiplicative identity of (L, +, ⊗ u , ) is u and hence we have the following Proposition 6.1.…”

Lattice-ordered power series fields

“…In the case of rooted groups, we also have the following. [5] Lattice-ordered power series fields 303 PROPOSITION [18] that every element of IW(>) is a locally inversely well-ordered subset of (A/A,, > ) , and hence there must exist n such that a k A t = a n A t for all k > n . But then A n a n A t contains the infinite set {a n , a n+l , a n+2 , ...}, which contradicts our choice of A e X(>).…”

“…Form the product n A T; for a € A and r e L, let a,r e]l A t be the vectors , {r a p), +, •), is a commutative ring with 7 = T as the multiplicative identity (cf. [4], [5], [13], [16], [20], [21], [22] [9], [13], [16], [17], [18], [19], [20], [21], [22]. When Ribenboim's convolution rings [20], [21], [22] are based on a group, they are a special case of this construction; he requires the supports to be locally inversely well-ordered and to have no infinite disjoint subsets.)…”

“…When Ribenboim's convolution rings [20], [21], [22] are based on a group, they are a special case of this construction; he requires the supports to be locally inversely well-ordered and to have no infinite disjoint subsets.) If ( (C) In [18], it was shown how to construct a lattice-ordered power series field over a torsion-free rooted abelian group. This process may be generalized to an arbitrary rooted abelian group (A, •, >) as follows.…”