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

Fields of quotients of lattice-ordered domains

“…Then y is a d-element and hence by Proposition 3.1, y = τg for 0 < τ ∈ T and g ∈ G. But by Corollary 2.4 of [17], y is in the maximal totally…”

Lattice-ordered Fields Determined by d-elements

“…It was also shown in [10] that if L is standard, then M(L) is the unique maximal totally ordered subfield of (L, +, ·, ). Note that all known -fields are standard.…”

“…It was observed in [10] that P(L) is the positive cone of a compatible partial order, , on L; it is easy to see that L ≥ ⊆ L and that P(L) = L = L ≥ if and only if 1 ∈ L ≥ .…”

“…By [10,Theorem 4.4], M(L) is a totally ordered subfield of (L, +, ·, ) if and only if F(L) is totally ordered by ; an -field satisfying either of these equivalent conditions is called standard. It was also shown in [10] that if L is standard, then M(L) is the unique maximal totally ordered subfield of (L, +, ·, ).…”

“…So B is a Wilson basis of L over T and hence L has a compatible lattice order, , such that B is an -basis of L over M(L) (see [14] and [11,Section 3.C]). By [10,Proposition 5.6] Proof Since each b i is a root, Theorem 6.2 implies that B ⊆ D(L). So by Theorem 4.8, (L, +, ·, ) is an -field with v -basis B, and by Corollary 5.4, L ≥ = aL for some a ∈ L ≥ .…”