A lattice-ordered ring is a ring equipped with a partial order that makes its additive group a lattice-ordered
group and has 
whenever 
and 
.
The order is translation invariant, and each pair of elements has a meet
and a join (Birkhoff and Pierce 1956). The order need not
be total.
Birkhoff and Pierce (1956) asked whether the usual complex numbers, regarded as a field, admits such an order.
He (2026) claims a construction, with at least continuum many distinct partial orders.
For every real transcendental 
, the claimed order satisfies 
, 
, and 
and restricts to the usual order on 
. The written construction contains an unresolved enumeration
step in Theorem 5.1. It replaces an enumeration of a countable algebraically closed
field by 
,
which does not enumerate that field. The claim therefore does not yet supply a verified
resolution here.
A compatible total order on the field is impossible because every nonzero square would be positive, whereas 
. That familiar obstruction does not by itself rule out
a compatible partial lattice order.
Bao and Zhang (2026) reported an independent construction of compatible lattice orders on 
,
with 
and 
incomparable with zero. Their extension procedure does not require a countable enumeration
of 
.
They obtain 
distinct positive cones as subsets of 
, where 
is the continuum. This counts
orders, rather than isomorphism classes. The full construction requires independent
assessment. Bao and Zhang (2026) report AI assistance in developing and checking
the extension argument.
." 27 Oct 2026.