Old and New Unsolved Problems in Lattice-Ordered Rings that need not be f-Rings

Fields of quotients of lattice-ordered domains

“…By Propositions 5.2 and 6.1, if R[Z] q had a 3-consistent archimedean lattice order, then either Ψ iw : R[Z] q −→ R((Z)) iw or Ψ w : R[Z] q −→ R((Z)) w would be a lattice homomorphism and hence the field of quotients of R[Z] in R((Z)) iw or R((Z)) w would be a sublattice. But Henriksen has shown in [10] (see also [11]) that Ψ iw (R[Z] q ) is not a sublattice of R((Z)) iw and Ψ w (R[Z] q ) is not a sublattice of R((Z)) w .…”

“…Of course, this leaves open the possibility that the field of quotients of R[x] may have a compatible lattice order that differs from that induced from R((x)). In [11], Henriksen revisited his example and asked again whether the field of fractions of R[x] can be lattice-ordered in a way that extends the usual order on R[x].…”

A class of archimedean lattice ordered algebras

“…Though certain other -algebras seem to be worthy of consideration, such as almost f -algebras, very little attention has been paid to -algebras that are not f -algebras. Very recently, M. Henriksen [11] expressed his wish to see more papers dealing with -algebras rather than falgebras. Besides, he furnished several strong reasons that should motivate workers in -algebras to start thinking beyond the f -algebras context.…”