A partially ordered group is an Abelian group with a partial ordering $\le$ and $g\le h$ implies $g+i \le h+i$. A lattice ordered group is a partially ordered group such that binary infima and suprema always exist and are well-defined.\n\n\ If $R$ is a commutative integral domain, $Q$ is its field of fractions, $Q^*$ is the multiplicative group of $Q$, and $U(R)$ is the group of units of $R$, the quotient group $Q^*/U(R)$ partially ordered by $aU \le bU$ whenever $a^{-1} b \in R$ is called the divisibility group of $R$.\n\n\ A valuation on a field $Q$ by a totally ordered group $G$ is a map $v$: $Q^* \to G$ such that $v(x+y)\ge \inf(v(x),v(y))$ if $x+y \not = 0$, and $v(xy)=v(x)+v(y)$.\n\n\ For any lattice ordered group $G$, there is a lattice embedding $f$: $G \hookrightarrow G' = \prod\limits_{M \in \Gamma} G_M$ into a direct product of totally ordered groups $G_M$ (with the product ordering); let $\pi_M: $G' \to G_M$ be the canonical projections. For the field $Q = k(\{Y_g: g \in G\})$, define a valuation $\phi_M$: $Q^* \to G_M$ as follows: for the monomials, $\phi_M(\c \prod\limits_{i=1}^r Y_{g_i}^{n_i}) = \sum\limits_{i=1}^r n_i \pi_M(f(g_i))$; for a polynomial, the valuation is defined to be the infimum of the valuations of its monomials; then naturally extend the map to quotients of polynomials. Now define $\prod\limits_{M \in \Gamma} \phi_M = \phi$: $Q^* \to G'$ and $R = \{0\} \cup \{x \in Q^*: \phi(x) \ge 0\}$. $R$ is the Jaffard-Ohm-Kaplansky construction of an integral domain with quotient field $Q$ and divisibility group $\phi(Q^*) = f(G) \cong G$.
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