The existence of the compatible lattice order on the complex field $\mathbb{C}$

In this paper, we give an affirmative answer to the open question posed by G.Birkhoff and R.S.Pierce in 1956 concerning the existence of a compatible lattice order on the complex field $\mathbb{C}$. Using valuation rings theory, basis extension in infinite-dimensional vector spaces and extension of field embeddings to algebraic extensions, we obtain that for every real $t>1$ transcendental over $\mathbb{Q}$, the compatible lattice order can be chosen such that $0\prec1$, $0\vee i=t$, and $1\wedge t=0$. Consequently, $\mathbb{C}$ admits at least $2^{\aleph_0}$ distinct compatible lattice orders, each extending the usual order on $\mathbb{Q}$.