an isomorphism of graded algebras $$A_{\mathbb{K}}^*({{\mathcal A}}) \cong H^*(M_{{{\mathcal A}}}, \mathbb{K}),$$ that is, $\mathcal L({{\mathcal A}})$ determines $H^*(M_{{{\mathcal A}}}, \mathbb{K})$ as a graded ring.

A similar object is the [*quadratic Orlik-Solomon algebra*]{} of a central arrangement ${{\mathcal A}}$, denoted $\overline{A^*_{{\mathbb{K}}}}({{\mathcal A}})$, defined as the quotient of $E^*({{\mathcal A}})$ by the ideal spanned by relations of type

$e_{H_{i}}e_{H_{j}}+e_{H_{j}}e_{H_{k}}=e_{H_{i}}e_{H_{k}}\, ,$

for $H_{i},H_{j},H_{k}$ such that $r(\{H_{i},H_{j},H_{k}\})=2$. For fiber-type arrangements, the Orlik-Solomon algebra is [*quadratic*]{}, i.e., $\overline{A^*_{{\mathbb{K}}}}({{\mathcal A}})=A^*_{{\mathbb{K}}}({{\mathcal A}})$; see Falk [@Fa1] and Shelton-Yuzvinsky [@SY].

The converse of Theorem \[thm:OS\] is however no longer true: the arrangements from Example 3.77 [@OT] have isomorphic cohomology algebras, but different lattices. This inconvenient disappears, when adding information provided by [*natural $1$-markings*]{}.

\[1mrk\] A [*$1$-marking*]{} of a space $X$ of finite type is a distinguished ordered ${\mathbb{Z}}$-basis of $H^1(X)$. A [*$1$-marking*]{} of a finitely generated group $G$ with free abelianization $G_{ab}$ is a distinguished ordered ${\mathbb{Z}}$-basis of $G_{ab}$. [*$1$-marked isomorphisms*]{} must preserve the distinguished ordered ${\mathbb{Z}}$-bases, in both cases.

Ordered hyperplane complements have a [*natural*]{} $1$-marking in cohomology, namely $\{ e_H \mid H\in {{\mathcal A}}\}$.

\[thm:1m\] For central arrangements, the cohomology ring $H^*(M_{{{\mathcal A}}})$, together with its natural $1$-marking, determines the lattice $\mathcal L({{\mathcal A}})$.

Now, let us 