# Temporal Multimodal Data-Processing Algorithms Based on Algebraic System of Aggregates
## 4 Basics Notions of ASA
\begin{array}{l}A_{1}=\llbracket M_{1},M_{2},\ M_{3}\ |\ \left\langle a_{1}^{1,1},a_{1}^{1, 2},a_{3}^{1,1}\right\rangle,\left\langle a_{1}^{1,2},a_{2}^{1,2}\right\rangle,\left\langle a_{1}^{1,3},a_{2}^{1,3},a_{3}^{1,3},a_{4}^{1,3}\right\rangle \llbracket\\ A_{2}=\llbracket M_{1},M_{2},\ M_{3}\ |\ \left\langle a_{1}^{2,1},a_{2}^{2 }\right\rangle,\left\langle a_{1}^{2,2},a_{2}^{2,2},a_{3}^{2,2}a_{4}^{2,2},a_ {5}^{2,2}\right\rangle,\left\langle a_{1}^{2,3},a_{2}^{2,3}\right\rangle \llbracket\end{array} \right.\end{array} \tag{3}\]  
In Figure 1, the color of an element represents data modality (elements of set \(M_{1}\) are blue, elements of set \(M_{2}\) are brown and elements of set \(M_{3}\) are green), the first value in the element's number designates the aggregate (\(A_{1}\) or \(A_{2}\)), and the second value in the element's number is an ordering number of the element in the tuple belonging to a certain set. For example, the blue circle, which contains the numbers **1-1**, represents the element \(a_{1}^{1,1}\) that belongs to the set \(M_{1}\) from the definition of \(A_{1}\) and the green circle, which contains the numbers **2-1**, represents the element \(a_{1}^{2,3}\) that belongs to the set \(M_{3}\) from the definition of \(A_{2}\). **Definition 3**. : _Aggregates \(A_{1}\) and \(A_{2}\) are called quasi-compatible (\(A_{1}\doteq A_{2}\)) if the type and sequence order of the sets in them partially coincide, while there is no requirement for the equality of the lengths of these aggregates, i.e., the conditions are fulfilled:_  
\[\left\{ \begin{array}{c}\{A_{1}\}\
cong\{A_{2}\}\\ \{A_{1}\}\cap\{A_{2}\}\
eq\varnothing.\end{array} \right. \tag{4}\]  
For example, the aggregates defined by (5) and illustrated by Figure 2 are quasi-compatible. \[ \begin{array}{c}A_{1}=\llbracket M_{1},M_{2},\ M_{3}^{1}\ \big{|}\ \left<a_{1}^{1,1},a_{2}^{1,1},a_{3}^{1,1} \right>,\left<a_{1}^{1,2},a_{2}^{1,2}\right>,\left<a_{1}^{1,3},a_{2}^{1,3},a_ {3}^{1,3},a_{4}^{1,3}\right>\rrbracket,\\ A_{2}=\llbracket M_{1},M_{2},\ M_{3}^{2}\ \big{|}\ \left<a_{1}^{21,2},a_{2}^{1,1} \right>,\left<a_{1}^{22,2},a_{2}^{2,2},a_{3}^{2,2},a_{4}^{2,2},a_{5}^{2} \right>,\left<a_{1}^{23,3},a_{2}^{2,3}\right>\rrbracket.\end{array}  \tag{5}\]  
**Definition 4**. : _Aggregates \(A_{1}\) and \(A_{2}\) are called incompatible (\(A_{1}\doteq A_{2}\)), if the type and sequence of the sets in them do not match, that is, the condition is fulfilled:_  
\[\{A_{1}\}\cap\{A_{2}\}=\varnothing. \tag{6}\]  
For example, the aggregates defined by (7) and illustrated by Figure 3 are incompatible. \[ \begin{array}{c}A_{1}=\llbracket M_{1}^{1},M_{2}^{1},\ M_{3}^{1}\ \big{|}\ \left<a_{1}^{1,1},a_{2}^{1,1},a_{3}^{1,1}\right>,\left<a_{1}^{1,2},a_{2}^{1,2 }\right>,\left<a_{1}^{1,3},a_{2}^{1,3},a_{4}^{1,3}\right>\rrbracket,\\ A_{2}=\llbracket M_{1}^{2},M_{2}^{2},\ M_{3}^{2}\ \big{|}\ \left<a_{1}^{2,1},a_{2}^{2,1} \right>,\left<a_{1}^{2,2},a_{2}^{2,2},a_{3}^{2,2},a_{4}^{2,2},a_{5}^{2} \right>,\left<a_{1}^{23,3},a_{2}^{2,3}\right>\rrbracket.\end{array}  \tag{7}\]  
A special case of incompatibility is hidden compatibility. Figure 1: An example of two compatible aggregates. Figure 3: An example of two incompatible aggregates. **Definition 5**. : _Aggregates \(A_{1}\) and \(A_{2}\) are called hiddenly compatible, \(A_{1}\)\((\doteq)\)\(A_{2}\), if both aggregates have the same set of sets, but their ordering is different, i.e., the conditions are fulfilled:_  
\[\left\{ \begin{array}{c}\{A_{1}\}\
ot\equiv\{A_{2}\}\\ \mid A_{1}\mid=\mid A_{2}\mid=N\\ \quad\forall M_{j}\subset\{A_{k}\},\end{array} \right. \tag{8}\]  
where \(j=[1,\ldots,N]\), \(k=[1,\ 2]\). For example, the aggregates defined by formulas (9) and illustrated by Figure 4 are hiddenly compatible. \[