# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 3 Use of the Theory of Quasi-Automata
The process of communication between cluster heads is described by the change of \(\vec{v}\) by means of multiplying \(\vec{v}\) by a square matrix of real numbers:  
\[I_{M}=\left\{\mathbf{A}_{k}= \begin{bmatrix}a_{11}&\ldots&a_{1n}\\ \ldots&\ldots&\ldots\\ a_{n1}&\ldots&a_{nn}\end{bmatrix} \mid a_{ij}\in\langle 0;1\rangle\text{ for }i\geq j,a_{ij}=0\text{ for }i<j;i,j\in\{1,\ldots,n\};\|\mathbf{A}_{k}\|_{1}\leq 1 \right\},\]  
where \(\|\mathbf{A}_{k}\|_{1}\) is the column norm of \(\mathbf{A}_{k}\). In other words, \(I_{M}\) is a subset of the set of upper triangular matrices, i.e., we can also write:  
\[I_{M}=\left\{\mathbf{A}_{k}= \begin{bmatrix}a_{11}&0&0&\ldots&0\\ a_{21}&a_{22}&0&\ldots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ a_{n1}&a_{n2}&a_{n3}&\ldots&a_{nn}\end{bmatrix} \mid\|\mathbf{A}_{k}\|_{1}\leq 1\right\} \tag{6}\]Now, these two sets \(S_{v}\) and \(I_{M}\) will be linked with a transition function \(\delta\) by:  
\[\delta(\mathbf{A},\vec{v})=\vec{v}\cdot\mathbf{A} \tag{7}\]  
for all \(\mathbf{A}\in I_{M}\) and all \(\vec{v}\in S_{v}\). If we regard the usual matrix multiplication (only swapped), i.e., \(\mathbf{A}\odot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}\) for all \(\mathbf{A},\mathbf{B}\in I_{M}\), then \((I_{M},\odot)\) is a monoid such that the free monoid \(I_{M}^{*}=I_{M}\). Thus we can regard the triple \((I_{M},S_{v},\delta)\) and study whether it is a quasi-automaton. In our context, the operation of creating words from input symbols of our alphabet \(I\) will be matrix multiplication, i.e., a word will be a product of matrices, i.e., again a matrix. **Theorem 1**. : _The triple \((I_{M},S_{v},\delta)\) is a quasi-automaton._  
Proof. : The identity matrix \(E_{n}\) is the neutral element of \(I_{M}\). Property 1 of Definition 4 holds trivially. Verification of Property 2 is also straightforward:  
\[\delta(\mathbf{A},\delta(\mathbf{B},\vec{v}))=\delta(\mathbf{A},\vec{v}\cdot \mathbf{B})=(\vec{v}\cdot\mathbf{B})\cdot\mathbf{A}=\vec{v}\cdot\mathbf{B} \cdot\mathbf{A}=\delta(\mathbf{B}\cdot\mathbf{A},\vec{v})=\delta(\mathbf{A} \odot\mathbf{B},\vec{v}).\]  
The set \(I_{M}\) consists of matrices such that the column norm \(||\mathbf{A}||_{1}\) is at most one. In the following Remark, we show that without this condition the set \(I_{M}\) would not be closed with respect to \"\(\odot\)\". **Remark 1**. : _Suppose two matrices \(\mathbf{A},\mathbf{B}\in I_{M}\). If we denote:_  
\[\mathbf{A}= \begin{bmatrix}a_{11}&0&0&\dots&0\\ a_{21}&a_{22}&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ a_{n1}&a_{n2}&a_{n3}&\dots&a_{nm}\end{bmatrix} \qquad\mathbf{B}= \begin{bmatrix} b_{11}&0&0&\dots&0\\ b_{21}&b_{22}&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ b_{n1}&b_{n2}&b_{n3}&\dots&b_{nm}\end{bmatrix} ,\]  
_then,_  
\[\mathbf{B}\odot\mathbf{A}=\mathbf{A}\cdot\mathbf{B}= \begin{bmatrix}\sum \limits_{i=1}^{1}a_{1i}b_{i1}&0&0&\dots&0\\ \sum\limits_{i=1}^{2}a_{2i}b_{i2}&\sum\limits_{i=2}^{2}a_{2i}b_{i2}&0&\dots&0 \\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \sum\limits_{i=1}^{n}a_{ni}b_{in}&\sum\limits_{i=2}^{n}a_{ni}b_{in}&\sum \limits_{i=3}^{n}a_{ni}b_{in}&\dots&\sum\limits_{i=n}^{n}a_{ni}b_{in}\end{bmatrix} \]  
_and it is obvious that the column norm \(\|\mathbf{B}\odot\mathbf{A}\|_{1}\) will not exceed \(1\), i.e., \(\mathbf{B}\odot\mathbf{A}\in I_{M}\). Indeed, suppose that \(\mathbf{A}\) is an all-ones matrix upper triangular matrix (which, of course violates the condition that \(\|\mathbf{A}\|_{1}\leq 1\)). Then all the sums in \(\mathbf{B}\odot\mathbf{A}\) reduce to \(\sum\limits_{i=j}^{n}b_{ij}\) which are (due to the fact that \(\|\mathbf{B}\|_{1}\leq 1\)) smaller than \(1\). If moreover \(\|\mathbf{A}\|_{1}\leq 1\), none of the sums becomes greater. Of course, in \(I_{M}\) we could have used the row norm instead of column one with the same result._  
**Example 1**. : _Regard sensor nodes deployed under water, which are divided into four tiers with tier 1 being \(0-25\)\(m\), tier 2 being \(25-50\)\(m\), tier 3 being \(50-75\)\(m\), and tier 4 being \(75-100\)\(m\) under water.