### A class of multiplicative lattices

We study the multiplicative lattices $L$ which satisfy the condition $a=(a:(a:b\left)\right)(a:b)$ for all $a,b\in L$. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $\mathbb{Z}$ or $\mathbb{R}$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.