## 2 Formalisation and Generalised Working Definition for Ordinal Class Structures
### Formalisation  
Let \(X_{\Omega}\) be a \(c\)-class classification task, which is defined by the \(d\)-dimensional data set \(X\subset\mathbb{R}^{d}\), \(d\in\mathbb{N}\), and the corresponding set of class labels \(\Omega=\{\omega_{1},\ldots,\omega_{c}\}\), with \(c>2\). We denote the resulting index set as \(I\), i.e., \(I=\{1,\ldots,c\}\). Each element of \(X_{\Omega}\) is a task-related object, which is a pair consisting of a data sample and its true class label, i.e., \(X_{\Omega}=\{(x_{i},y_{i})\}_{i=1}^{N},y_{i}\in\Omega\), \(\forall i=1,\ldots,N\), whereby \(N=|X_{\Omega}|\) denotes the number of elements in the set \(X_{\Omega}\). Specifically, it holds, \(X_{\Omega}\subsetneq X\times\Omega\). Moreover, by \(X_{\Omega}^{ij}\), we denote the binary subtask that is restricted to the classes \(\omega_{i}\) and \(\omega_{j}\), i.e., for all \(i,j\in I\), with \(i\
eq j\), we define  
\[X_{\Omega}^{ij}:=\{(x,y)\in X_{\Omega}|y=\omega_{i}\;\vee\;y=\omega_{j}\}. \tag{1}\]  
Therefore, it holds that \(X_{\Omega}^{ij}=X_{\Omega}^{ji,i}\;\forall i,j\in I,\,i\
eq j\). For the definition of ordinal classification tasks, in this work, we introduce the term _level of separability measures_, which we define as follows, in Definition 1. Figure 1: General classification task processing steps. **Left**: Sequential processing steps. **Right**: Step-specific processing examples. The detection of ordinal class structures is included in the Data Analysis step (highlighted in green colour, in the online version of the manuscript). **Definition 1** (Level of Separability Measures). : _Let \(X_{Y}\), \(X\subset\mathbb{R}^{d}\), \(d\in\mathbb{N}\), and \(Y=\{0,1\}\) constitute a binary classification task in a \(d\)-dimensional feature space. Furthermore, let \(X_{\mathcal{Y}}\) be the corresponding binary classification task, where we interchange the class labels of all samples from the task \(X_{Y}\), i.e., \(X_{\mathcal{Y}}:=\{(x,1-y)|(x,y)\in X_{\mathcal{Y}}\}\). Additionally, for each \(X\subset\mathbb{R}^{d}\), we define the set \(X^{*}_{Y}\) as \(X^{*}_{Y}=X\times Y\). We denote the non-constant and non-random mapping \(\mu\) as a level of separability measure (LSM), if \(\mu\) fulfils the following properties:_  
\[ \begin{array}{llll}\text{(P0)}&\mu(X_{Y})\geq 0,&\forall X_{Y}\subset \mathbb{R}^{d}\times\{0,1\},&\text{(nonnegative)},\\ \text{(P1)}&\mu(X^{*}_{Y})\leq\mu(Z_{Y}),&\forall Z_{Y}\subset\mathbb{R}^{d} \times\{0,1\},&X\subset\mathbb{R}^{d},&\text{(point-separating)},\\ \text{(P2)}&\mu(X_{Y})=\mu(X_{\mathcal{Y}}),&\forall X_{Y}\subset\mathbb{R}^{d} \times\{0,1\},&\text{(label invariant)}.\end{array}  \tag{2}\]  
Note that a higher value for \(\mu\) implies a higher level of separability. The properties defined in Equation (2) are further discussed in the following remark, i.e., in Remark 1. **Remark 1** (Properties of LSM mappings). : _Let \(\mu\) be an LSM mapping by Definition 1. Furthermore, let \(X_{\{0,1\}}\subset\mathbb{R}^{d}\times\{0,1\}\) constitute a binary classification task. Property (P1) of Equation (2) implies that if we set \(Z=X^{*}_{\{0,1\}}=X\times\{0,1\}\), then the value of \(\mu(Z)\) is equal to the minimum value of \(\mu\), across all \(X_{\{0,1\}}\subset\mathbb{R}^{d}\times\{0,1\}\). Therefore, we say that \(\mu\) is point-separating. More precisely, the set \(X^{*}_{\{0,1\}}\) is defined as \(X^{*}_{\{0,1\}}=X\times\{0\}\cup X\times\{1\}\), i.e., \(X^{*}_{\{0,1\}}=\{(x_{1},0),(x_{1},1),\ldots,(x_{N},0),(x_{N},1)\}\). In such a set, each data point is assigned to both of the class labels, leading to the lowest level of separability, in combination with any LSM mapping \(\mu\). Property (P2) of Equation (2) simply implies that interchanging the labels of all data points \(x\in X\) does not change the value of \(\mu\), evaluated in combination with the set \(X\).