# Elements of Hyperstructure Theory in UWSN Design and Data Aggregation
## 3 Use of the Theory of Quasi-Automata  
In Definition 2, the concept of quasi-order hypergroup is defined. [PERSON] and [PERSON] [25] relate these to the theory of quasi-automata, i.e., automata without output. For an automaton they construct a quasi-order hypergroup of its state set and show that the automaton is connected if and only if the state hypergroup is inner irreducible as well as strogly connected, i.e., we can reach any state from any other state, if and only if the state hypergroup is (in a special way) cyclic. In other words, if we look at the problem of data aggregation from the point of view of the automata theory,where every step is an application of the transition function with the initial state \"data aggregation to begin\" and the desirable state \"data from all elements collected\" (or rather \"useful data from all elements sent\" since every \(CH\) not only receives data but also separates useful data from useless ones), we should be interested in constructing such automata or studying their properties. We call the concept defined below quasi-automaton even though this term is not much frequent (we do this to be consistent with some earlier papers on hyperstructure theory). In fact, we could speak of semiautomata or deterministic finite automata (the below mentioned paper [PERSON] (2015) uses a general term automaton; however, notice that [PERSON] (2015), p. 107, plain text, defines automaton in the way of Definition 4, which is a definition adopted by the authors of [PERSON] (2015) in later years). For an overall discussion of the concepts and the reasons for our choice of the name see [PERSON] et al. [PERSON] et al. (2016). For some further reading and applications see also [PERSON] et al. [PERSON] et al. (2016); [PERSON] et al. (2017); [PERSON] et al. (2018). Definition 4: By a quasi-automaton we mean a structure \(\mathbb{A}=(I,S,\delta)\) such that \(I\
eq\emptyset\) is a monoid, \(S\
eq\emptyset\) and \(\delta:I\times S\to S\) satisfies the following condition:  
1. There exists an element \(e\in I\) such that \(\delta(e,s)=s\) for any state \(s\in S\);
2. \(\delta(y,\delta(x,s))=\delta(xy,s)\) for any pair \(x,y\in I\) and any state \(s\in S\). The set \(I\) is called the input set or input alphabet, the set \(S\) is called the state set and the mapping \(\delta\) is called next-state or transition function. Condition 2 is called GMAC (Generalized Mixed Associativity Condition). In [PERSON] (2015), [PERSON] and [PERSON] defined what they called a state hypergroup of an automaton. This is in fact a state set with a special hyperoperation, defined by means of the transition function. In this way, the concept of a state hypergroup is fixed to the automata theory. However, the way of defining this concept is a parallel to the concept of quasi-order hypergroups, which means that state hypergroups of quasi-automata are quasi-order hypergroups. The fact that the below defined \((S,\circ)\) is a hypergroup, or rather quasi-order hypergroup, (hence the name state hypergroup) was proved in [PERSON] (2015). (Notice that in [PERSON] (2015)\(I\) and \(S\) are swapped.) Definition 5: Let \(\mathbb{A}=(I,S,\delta)\) be an automaton. We define a binary hyperoperation \"\(\circ\)\" on the state set \(S\) by:  
\[s\circ t=\delta(I^{*},s)\cup\delta(I^{*},t) \tag{3}\]  
for any pair of states \(s,t\in S\), where \(A^{*}\) is a free monoid of words over the (non-empty) alphabet \(A\). The hyperstructure \((S,\circ)\) is called state hypergroup of the automaton \(\mathbb{A}\). Some properties of automata following from properties of its state hypergroup are proved in Chvalinova (2015). This includes the properties of being connected or separated. Definition 6: Let \(\mathbb{A}=(I,S,\delta)\) be a quasi-automaton. A quasi-automaton \(\mathbb{B}=(I,S_{1},\delta_{1})\) such that \(S_{1}\subseteq S\) and \(\delta_{1}\) is a restriction of \(\delta\) on \(I\times S_{1}\) and \(\delta(a,s)\in S_{1}\) for any state \(s\in S_{1}\) and any word \(a\in I^{*}\), is called a sub quasi-automaton of \(\mathbb{A}\).