## 2 Formalisation and Generalised Working Definition for Ordinal Class Structures
### Formalisation
Therefore, we say that \(\mu\) is label invariant, or simply symmetric._  
By \(\mathcal{M}^{d}\), we denote the set of all mappings that measure the level of separability of any binary classification task from the \(d\)-dimensional feature space, i.e.,  
\[\mathcal{M}^{d}:=\{\mu|\mu\text{ is an LSM mapping in }\mathbb{R}^{d}\times\{0,1\}\text{ by Definition 1}\}. \tag{3}\]  
Note that by Definition 1, each element of the set \(\mathcal{M}^{d},d\in\mathbb{N}\), not only fulfils the properties (P0), (P1), and (P2) of Equation (2), but is also a _non-constant_ and _non-random_ mapping. Moreover, note that the set \(\mathcal{M}^{d}\) is non-empty, for all \(d\in\mathbb{N}\), as we briefly discuss in the following example, i.e., Example 1. **Example 1** (Existence of LSM mappings). : _Note that the set \(\mathcal{M}^{d}\) is non-empty, for all \(d\in\mathbb{N}\). Let CM be a deterministic classification model, e.g., a support vector machine, which is trained based on the set \(X_{\{0,1\}}\subset\mathbb{R}^{d}\times\{0,1\}\). Let \(err_{CM}\in[0,1]\) be the resubstitution error (training error) of classifier CM, i.e., \(err_{CM}\) is the fraction of data points in \(X\) that are misclassified by classifier CM. Then, \(\mu:X_{\{0,1\}}\mapsto C-err_{CM}\) fulfils the properties of Equation (2), for each absolute term \(C\geq 1\), and any deterministic classification model CM._  
Let \(\mu\in\mathcal{M}^{d}\) be an LSM mapping. For all \(i,j\in I\), we define \(\mu_{i,j}\in\mathbb{R}\) as follows:  
\[\mu_{i,j}:= \begin{cases}\mu\Big{(}X^{i,j}_{\Omega}\Big{)},&\text{if }i\
eq j,\\ 0,&\text{if }i=j,\end{cases}  \tag{4}\]  
which measures the level of separability between the samples from class \(\omega_{i}\) and the samples from class \(\omega_{j}\). Therefore, for \(i,j,k,l\in I\), the statement, \(\mu_{i,j}>\mu_{k,l}\), implies that it is _easier_ to separate the classes \(\omega_{i}\) and \(\omega_{j}\) from each other, than to separate the classes \(\omega_{k}\) and \(\omega_{l}\) from each other. Thus, if it holds, \(\mu_{i,j}>\mu_{k,l}\), we simply say that the binary classification task \(X^{ij}_{\Omega}\) has a _higher_ level of separability than the binary classification task \(X^{kj}_{\Omega}\). Note that from Equations (1) and (4), it directly follows that \(\mu_{i,j}=\mu_{j,j},\forall i,j\in I\). In addition, note thatsetting \(\mu_{i,j}\) to zero, for all \(i\in I\), is a logical consequence, as it is not possible to separate two _identical_ data sets from each other. Furthermore, let \(\mathcal{T}^{c}\) be the set of all permutations of the set \(I\). More precisely, each \(\tau\in\mathcal{T}^{c},\tau:\{1,\ldots,c\}\to\{1,\ldots,c\}\), is a bijective function. In addition, by \(-\tau\in\mathcal{T}^{c}\), we denote the _reversed_ permutation of \(\tau\in\mathcal{T}^{c}\). For instance, for the _identity_ permutation, \(id:(1,\ldots,c)\mapsto(1,\ldots,c)\), it holds, \(-id:(1,\ldots,c)\mapsto(c,c-1,\ldots,1)\). By \(M_{(X_{\Omega,\mu})}\in\mathbb{R}_{\geq 0}^{c\times c}\), we denote the _pairwise separability matrix_ (PSM), consisting of the elements \(\mu_{i,j}\), i.e., \(M_{(X_{\Omega,\mu})}:=(\mu_{i,j})_{i,j=1}^{c}\). Moreover, by \(M_{(X_{\Omega,\mu})}^{(\tau)}\), we define the PSM whose rows and columns are rearranged specific to permutation \(\tau\in\mathcal{T}^{c}\), i.e., \(M_{(X_{\Omega,\mu})}^{(\tau)}:=(\mu_{\tau(i),\tau(j)})_{i,j=1}^{c}\). For reasons of simplicity, we will denote \(M_{(X_{\Omega,\mu})}^{(\tau)}\) simply by \(M^{(\tau)}\), with \(M=M^{(id)}\), whereby \(id\) denotes the identity permutation. More precisely, \(M^{(\tau)}\) can be depicted as follows:  
\[M^{(\tau)}= \begin{pmatrix}0&\mu_{\tau(1),\tau(2)}&\ldots&\mu_{\tau(1),\tau(c)} \\ \mu_{\tau(2),\tau(1)}&0&\ldots&\mu_{\tau(2),\tau(c)}\\ \vdots&\vdots&\ddots&\vdots\\ \mu_{\tau(c),\tau(1)}&\mu_{\tau(c),\tau(2)}&\ldots&0\end{pmatrix} . \tag{5}\]  
Note that by definition, matrix \(M^{(\tau)}\) is symmetric for all \(\tau\in\mathcal{T}^{c}\).