c  <  b$.

Let $O\subseteq \{<_1, <_2, \ldots, \} \cup \{{\ensuremath{{S}}}_1,{\ensuremath{{S}}}_2,\ldots, \}$ where the $<_{i}$ and ${\ensuremath{{S}}}_{i}$ are binary relation symbols. An $O$-structure ${\ensuremath{\mathfrak{A}}}$ is *ordered* if each relation $<_{i}^{{\ensuremath{\mathfrak{A}}}}$ is a linear order and each relation ${\ensuremath{{S}}}_{i}^{{\ensuremath{\mathfrak{A}}}}$ is the induced successor relation of a linear order. Furthermore, if ${\ensuremath{{S}}}_i\in O$ and $<_{i} \in O$, then ${\ensuremath{{S}}}_{i}^{{\ensuremath{\mathfrak{A}}}}$ is the induced successor relation of $<_{i}^{{\ensuremath{\mathfrak{A}}}}$. For convenience we often identify relation symbols with their respective interpretations. We use the symbols ${\ensuremath{{S}}}$ and $<$ if an ordered structure only has one linear order.

We say that two distinct elements $a$ and $b$ of ${\ensuremath{\mathfrak{A}}}$ are *$<_i$-close* if ${\ensuremath{{S}}}_i(a,b)$ or ${\ensuremath{{S}}}_i(b,a)$. Otherwise, $a$ and $b$ are *$<_i$-remote*. We say that $a$ and $b$ are *close* if they are $<_i$-close for some ${\ensuremath{{S}}}_i \in O$. Otherwise, $a$ and $b$ are *remote*. If the signature $O$ contains ${\ensuremath{{S}}}_i$ but not $<_i$ we also say ${\ensuremath{{S}}}_i$-close and so on.

For an ordered structure ${\ensuremath{\mathfrak{A}}}$ with two underlying linear orders $<_1$ and $<_2$ it will often be convenient to think of ${\ensuremath{\mathfrak{A}}}$ as a point set in the two-dimensional plane where $<_1$ orders points along the $x$-axis and $<_2$ orders points along the $y$-axis (see Figure \[figure:example:PointSetDataWord\]). 