# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 2 Mathematical Model
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and \(CH_{k}\) is such a cluster head that \(CH_{i}\leq CH_{k}\), \(CH_{j}\leq CH_{k}\), where \(a\leq b\) is a relation between elements of \(H\) such that: (1) \(s\leq s,s\leq CH_{i}\) and \(CH_{i}\leq s\) for all clusters \(cl_{i}\), (2) within the same cluster \(cl_{i}\) we have \(a_{j}\leq CH_{i}\) for all \(a_{j}\in cl_{i}\) while mutually different ordinary elements of the cluster are incomparable, (3) between clusters for \(a=CH_{i}\), \(b=CH_{j}\) the fact that \(a\leq b\) means that the tier of \(b\) (measured towards the surface) is smaller than or equal to the tier of \(a\), and (4) in all other cases \(a\) and \(b\) are not related. By \(CH_{k}\) above we mean a cluster head on the closest tier above both \(CH_{i}\) and \(CH_{j}\). Of course, \(CH_{k}\) always exists yet need not be unique as there may be more cluster heads at this closest tier. In such a case, we choose the most suitable one or regard all cluster heads as equal. Notice that, in our definitions, the fact that \(CH_{i}\leq CH_{j}\) and simultaneously \(CH_{j}\leq CH_{i}\) does not mean that \(CH_{i}=CH_{j}\), rather it only means that \(CH_{i}\) and \(CH_{j}\) are on the same tier. If we are able to chose the most suitable cluster head (further on we remark that we are), the relation \"\(\leq\)\" (restricted to \(H\setminus\{s\}\)) becomes partial ordering and we can write \(CH_{k}=\sup\{CH_{i},CH_{j}\}\) (with respect to the relation \"\(\leq\)\"). Finally, the element \(s\) is an element of \(H\) reserved for situations when \(a\) and \(b\) fail to communicate. It is artificially added to our set of elements \(H\) or we can agree that one (given the actual sensor deployment is of course carefully chosen) of elements of \(H\) will be \(s\). In this way, technically speaking, we should  
Figure 2: Cluster based approach to UWSN data aggregation—idealized deployment. The tiers need not be horizontal, we usually regard distance towards sink instead of depth. in fact write \(H_{e}=H\cup\{s\}\), where \(H_{e}\) could mean \"expanded\". Of course, if we choose the option of \(s\in H\), then \(H_{e}=H\). Under these definitions, \(a\cdot b\) is the element in which the data from \(a\) and \(b\) meet, and \(a*b\) is the path in which the data from both \(a\) and \(b\) can spread. The facts that \(a\cdot b=s\) or \(a*b=\{a,b\}\) or \(a*b=\{a,b,s\}\) all stand for communication failure. **Lemma 1**. : [1]__\((H,\leq)\) _is a quasi-ordered set._  
Suppose now that we have arbitrary \(a,b\in H\). Since the result of \(a\cdot b\) is such an element of \(H\) in which the data from \(a\) and \(b\) meet, it is natural to suppose that \(a\cdot b=b\cdot a\), i.e., that \"\(\cdot\)\" is commutative. However, we can suppose this only on condition that there exists such an algorithms that \(a\cdot b=CH_{k}=CH_{l}=b\cdot a\) for arbitrary clusters \(cl_{k},cl_{l}\). Further on suppose that such an algorithm exists, i.e., that \((H,\cdot)\) is a commutative groupoid. The following lemma is obvious. **Lemma 2**. : [1] _If \((H,\cdot)\) is a commutative groupoid, then \((H,*)\) is a commutative hypergroupoid._  
In the following lemma notice that weak associativity of the hyperoperation is defined as \(a*(b*c)\cap(a*b)*c\
eq\emptyset\) for all \(a,b,c\in H\); a quasi-hypergroup is a reproductive hypergroupoid. **Lemma 3**. : [1] _The hypergroupoid \((H,*)\) is a \(H_{v}\)-group, i.e., a weak associative quasi-hypergroup._  
**Lemma 4**. : [1] _The quasi-ordering \"\(\leq\)\" and the operation \"\(\cdot\)\" are compatible, i.e., for all \(a,b\in H\) such that \(a\leq b\) and an arbitrary \(c\in H\) there is \(a\cdot c\leq b\cdot c\) and \(c\cdot a\leq c\cdot b\)._  
Now, denote \(H_{CH}\subseteq H\) the set of cluster heads. This notation enables us to regard both clustering based systems and multipath systems because the fact that \(H_{CH}=H\) means that every element of \(H\) is a cluster head, i.e., the system is in fact a multipath one. In such a case the model simplifies substantially.