# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 3 Use of the Theory of Quasi-Automata
\mid c_{ii}\in\langle a_{ii} \cdot b_{ii};1\rangle\text{ and }c_{ij}\in\langle a_{ij}\cdot b_{ij};a_{ii}\cdot b_{ii}\rangle \text{ for }i\
eq j\right\} \tag{8}\]for all \(\mathbf{A},\mathbf{B}\in\hat{I}_{M}\), where \"\(\cdot\)\" is the usual multiplication of real numbers and \(c_{ij}\) belongs to a closed interval bounded by the product \(a_{ij}\cdot b_{ij}\) and either \(1\) or the product of the corresponding diagonal elements. **Remark 2**: _Notice that by removing the assumption regarding the column norm we are making our model more real-life because we regard memory losses and overflows. Indeed, suppose we have a vector e.g., \(\vec{v}=(0.8;0.7)\) and a matrix e.g., \(\mathbf{A}= \begin{bmatrix}1&0\\ 1&1\end{bmatrix} \). Then \(\delta(\mathbf{A},\vec{v})=\vec{v}\mathbf{A}=(1.5;0.7)\), which (since vector components are taken from the interval \(\langle 0;1\rangle\) because they describe percentages of memory capacity) must be reduced to \((1;0.7)\)._  
**Theorem 2**: _The pair \((\hat{I}_{M},*)\) is a commutative hypergroup._  
Commutativity of the hyperoperation \"\(*\)\" is obvious because we work with entry-wise multiplication of real numbers. For the test of associativity, i.e., \(\mathbf{A}*(\mathbf{B}*\mathbf{C})=(\mathbf{A}*\mathbf{B})*\mathbf{C}\) for all triples \(\mathbf{A},\mathbf{B},\mathbf{C}\in\hat{I}_{M}\), recall that we will be comparing sets of matrices of the same dimension, elements of which fall within intervals defined by Equation (8). As far as diagonal elements are concerned, there is obviously no problem because we make use of the multiplication of real numbers. For the non-zero, non-diagonal elements the lower bounds of the intervals are the products of their smallest elements, i.e., \(a_{ij}(b_{ij}c_{ij})=(a_{ij}b_{ij})c_{ij}\), while the upper bounds of the intervals are products of their greatest possible, i.e., diagonal, elements i.e., \(a_{ii}(b_{ii}c_{ii})=(a_{ii}b_{ii})c_{ii}\). Thus, in all cases, associativity of the hyperoperation follows from the associativity of (positive) real numbers (smaller than \(1\)). Reproductive axiom is also obviously valid because since the null matrix \(\mathbf{0}\) is an element of \(\hat{I}_{M}\) and \(\mathbf{A}*\mathbf{0}=\hat{I}_{M}\). Notice that in the following theorem, \(\delta(\mathbf{A},\vec{v})\) is the usual vector and matrix multiplication with the additional condition that no diagonal component is allowed to exceed \(1\). For its motivation see Remark 2. **Theorem 3**: _The triple \((\hat{I}_{M},S_{v},\delta)\) is a quasi-multiautomaton, where, for all \(\mathbf{A}\in\hat{I}_{M}\) and all \(\vec{v}\in S_{v}\),_  
\[\delta(\mathbf{A},\vec{v})=\left(\sum\limits_{i=1}^{n}v_{i}a_{i1},\ldots,\sum \limits_{i=1}^{n}v_{i}a_{in}\right), \tag{9}\]  
_where in case that \(\sum\limits_{i=1}^{n}v_{i}a_{ik}>1\) for some \(k\in\{1,2,\ldots,n\}\), we set, by default, \(\sum\limits_{i=1}^{n}v_{i}a_{ik}=1\)._  
Thanks to Theorem 2, we shall verify condition Equation (4) only. The left-hand side of the condition is:  
\[\delta(\mathbf{A},\delta(\mathbf{B},\vec{v}))=\delta(\mathbf{A},\vec{v}\cdot \mathbf{B})=\delta\left(\mathbf{A},\left(\sum\limits_{i=1}^{n}v_{i}b_{i1}, \ldots,\sum\limits_{i=1}^{n}v_{i}b_{in}\right)\right)=\left(\sum\limits_{i=1} ^{n}v_{i}b_{i1},\ldots,\sum\limits_{i=1}^{n}v_{i}b_{in}\right)\cdot \begin{bmatrix} a_{11}&\ldots&a_{1n}\\ \ldots&\ldots&\ldots\\ a_{n1}&\ldots&a_{mn}\end{bmatrix}