Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

symmetry S S Article Elements of Hyperstructure Theory in UWSN Design and Data Aggregation Michal Novák 1,* , Štepán Kˇrehlík 2and Kyriakos Ovaliadis 3 1Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic 2Department of Applied Mathematics and Computer Science, Masaryk University, Lipová 41a, 602 00 Brno, Czech Republic; [email protected] 3 Department of Electrical Engineering, Eastern Macedonia and Thrace Institute of Technology, Agios Loukas, 654 04 Kavala, Greece; [email protected] *Correspondence: [email protected].cz; Tel.: +420-54114-6077 Received: 8 May 2019; Accepted: 27 May 2019; Published: 29 May 2019   Abstract: In our paper we discuss how elements of algebraic hyperstructure theory can be used in the context of underwater wireless sensor networks (UWSN). We present a mathematical model which makes use of the fact that when deploying nodes or operating the network we, from the mathematical point of view, regard an operation (or a hyperoperation) and a binary relation. In this part of the paper we relate our context to already existing topics of the algebraic hyperstructure theory such as quasi-order hypergroups, EL -hyperstructures, or ordered hyperstructures. Furthermore, we make use of the theory of quasi-automata (or rather, semiautomata) to relate the process of UWSN data aggregation to the existing algebraic theory of quasi-automata and their hyperstructure generalization. We show that the process of data aggregation can be seen as an automaton, or rather its hyperstructure generalization, with states representing stages of the data aggregation process of cluster protocols and describing available/used memory capacity of the network. Keywords: clustering protocols; quasi-automaton; quasi-multiautomaton; semihypergroup; UWSN 1. Introduction Underwater wireless sensor networks (UWSN) are often used in environment monitoring where they review how human activities affect marine ecosystems, undersea explorations such as detecting oilfields, for disaster prevention, e.g., when monitoring ocean currents, in assisted navigation for the location of dangerous rocks in shallow waters, or for disturbed tactical surveillance for intrusion detection. The fact that such wireless sensor networks are deployed underwater results in profound differences from terrestrial wireless sensor networks. The key aspects that are different include the communication method, i.e., radio waves vs acoustic signals, cost (while terrestrial networks experience decreasing prices of components, underwater sensors are still expensive devices), memory capacity (because water is a problematic medium resulting in the loss of large quantities of data), power limitations due to the nature of the signal and longer distances handled, as well as problems related to the deployment of the network, i.e., issues connected to static or dynamic deployment. In underwater sensor networks, we commonly face challenges of limited bandwith, high bit error rates, large propagation delays, and limited battery resources caused by the fact that in an underwater environment, sensor batteries are impossible to recharge especially because no solar energy is available underwater. The power losses, which cannot be avoided, result in the need to reconfigure the network topology in order to maintain network connectivity and communication between sensor nodes. Thus, Symmetry 2019,11, 734; doi:10.3390/sym11060734 www.mdpi.com/journal/symmetry Symmetry 2019,11, 734 2 of 16 size of the UWSN coverage area and efficiency of data aggregation are affected. Obviously, efficiency in battery use influences network lifetime without sacrificing system performances. These differences are shown in Table 1. Table 1. Comparison of some features of terrestrial and underwater wireless sensor networks (UWSN). (Terrestrial) WSN UWSN Communication Media RF Waves Acoustic Waves Frequency High Low Node size Small Large Deployment Dense Sparse Power Low High Energy consumption Low High Propagation delay Low High Bandwidth High Low Path loss Low High Cost Inexpensive Expensive Memory Sensor nodes have low capacity Sensor nodes require large capacity We use different protocols for discovering and maintaining routes between sensor nodes. As mentioned in Novák, Kˇrehlík, and Ovaliadis [ 1 ], the most commonly used routing protocols are: Flooding, multipath, cluster, and miscellaneous protocols, see Wahid and Dongkyun [ 2 ]. In the flooding approach, the transmitters send a packet to all nodes within the transmission range. In the multipath approach, source sensor nodes establish more than one path towards sink nodes on the surface. Finally, in the clustering approach the sensor nodes are grouped together in a cluster. For an easy-to-follow reading on how UWSN’s work and on advantages of clustering see Domingo and Prior [3], the basic idea is shown in Figures 1,2. Recent research shows that the cluster based protocols give a great contribution towards the concept of energy efficient networks, see Ayaz et al. [ 4 ], Ovaliadis and Savage [ 5 ], or Rault, Abdelmadjid, and Yacine [ 6 ]. A common cluster based network consists of a centralized station deployed at the surface of the sea called a sink (or surface station) and sensor nodes deployed at various tiers inside the sea environment. These are grouped into clusters. In this architecture, each cluster has a head sensor node called a cluster head ( CH ). The cluster head is assumed to be inside the transmission range of all sensor nodes that belong to its cluster. Every cluster head operates as a coordinator for its cluster, performing significant tasks such as cluster maintenance, transmission arrangements, data aggregation, and data routing (Figure 2). Mathematical Background of the Model In the UWSN topology, several aspects are important for successful data aggregation. First of all, there must exist a path linking every element of the network to the surface station. However, these paths need not be unique as there might be multiple possible paths which the data from a given element can use to reach the surface station. Second, there always exists a cetain kind of ordering of the set of the network elements. They can be ordered with respect to their physical depth, with respect to their importance, with respect to communication priority, remaining battery power, etc. Finally, as data are collected, they are combined in the "upwards" elements in order to be sent further on. Thus one may employ techniques of algebra or graph theory in the description of the data aggregation process as has been recently done by Aboyamita et al., Domingo, or Jiang et al. [ 7 – 9 ]. However, given the multivalued nature of data aggregation (multiple paths, more than one possible links of elements, etc.), it seems relevant to make use of the elements of the algebraic hyperstructure theory. Notice that while in "classical" algebra, we regard operations, i.e., mappings f:Hn→H , in the algebraic hyperstructure theory we work with hyperoperations, i.e., mappings g:Hn→ P∗(H) , where P∗(H) is the power set of H with ∅ excluded (one need not consider this exclusion though). For Symmetry 2019,11, 734 3 of 16 the general introduction to the theory as well as definitions of concepts not explicitly defined further on, see Corsini and Leoreanu [10]. In the algebraic hyperstructure theory, there are several concepts which make use of the aspect of ordering. A small selection includes Comer, Corsini, Cristea, De Salvo et al. [ 11 – 15 ]. Further on we discuss three of these: EL –hyperstructures, quasi-order hypergroups, and ordered hyperstructures. Each of these concepts uses somewhat different background and assumptions: EL–hyperstructures are constructed from preand partially-ordered semigroups, i.e., the hyperoperation is defined using an operation and a relation compatible with it; Quasi-order hypergroups are constructed from pre-ordered sets, i.e., the hyperoperation is defined using a relation only; Ordered hyperstructures are algebraic hyperstructures on which a relation compatible with the hyperoperation is defined. All of these have been studied in depth and numerous results have been achieved in their respective theories. The idea of EL –hyperstructures has been implicitely present in a number of works since at least the 1960s, for example Pickett [ 16 ]. The definition and first results were given by Chvalina [ 17 ] and the theory has been elaborated by Novák (later jointly with Chvalina, Kˇrehlík, and Cristea) in a series of papers including [ 18 – 22 ]. It is to be noted that, since the class of EL –hyperstructures is rather broad, the aim of many theorems included in some of those papers was to establish a common ground for some already existing ad hoc derived results. Recently, some examples concerning various types of cyclicity in hypergroups have been constructed using EL –hyperstructures, see Novák, Kˇrehlík and Cristea [23]. The idea of quasi-order hypergroups was proposed by Chvalina in [ 17 , 24 , 25 ]. Some results achieved with the help of this concept are included in Corsini and Leoreanu [ 10 ]. Not to be missed are results concerning the theory of automata collected in Chvalina and Chvalinová [ 25 ]. It should be stressed that these results were motivated by Comer [26] and Massouros and Mittas [27]. Ordered hyperstructures were introduced by Heidari and Davvaz [ 28 ]. Numerous results have been published since, mainly by Iranian authors. For the following set of basic definitions see Novák, Kˇrehlík, and Ovaliadis [1]. Definition 1. By an EL –semihypergroup we mean a semihypergroup, in which, for all a , b∈H , there is a∗b={x∈H|a·b≤x}, where (H,·,≤)is a quasi-ordered semigroup. Proposition 1. [ 20 , 22 ] If, for all a , b∈H , there is {a , b} ∈ a∗b , then the EL –semihypergroup (H , ∗) is a hypergroup. If (H,·,≤)is a partially ordered group, then its EL–hypergroup (H,∗)is a join space. Definition 2. Let (H , ∗) be a hypergroupoid. We say that H is a quasi-order hypergroup, i.e., a hypergroup determined by a quasi-order, if, for all a , b∈H , a∈a3=a2 , and a∗b=a2∪b2 . Moreover, if a2=b2⇒a=b holds for all a,b∈H, then (H,∗)is called an order hypergroup. Proposition 2. [ 10 ] A hypergroupoid is a quasi-order hypergroup if and only if there exists a quasi-order " ≤ " on the set H such that, for all a,b∈H, there is a ∗b= [a)≤∪[b)≤. Definition 3. An ordered semihypergroup (H , ∗ , ) is a semihypergroup (H , ∗) together with a partial ordering "  " which is compatible with the hyperoperation, i.e., xy⇒a∗xa∗y and x ∗ay∗a for all a,x,y∈H. By a ∗xa∗y we mean that for every c ∈a∗x there exists d ∈a∗y such that c d. Notation. Further on, for some a∈H , by [a)≤ means the set {x∈H|a≤x} . For this reason, closed intervals will not be denoted by [a,b]but by ha;bi. Symmetry 2019,11, 734 4 of 16 2. Mathematical Model The mathematical model presented in this section was published as an extended abstract of the conference contribution Novák, Ovaliadis, and Kˇrehlík [ 1 ] presented by the authors of this paper at International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2017). UWSNs consist of elements of different types: First, we have surface stations, which pass data to a ship or to a data-collecting station located on the sea shore; second, we have sensor nodes deployed at various tiers in water or at the sea bed. The sensors, which are deployed in water, can function as sensors measuring the requested data or as transporters of information from seabed sensors. In any case, information collected from all sensors must be passed to surface stations. From these it can be collected either by a ship passing by or, alternatively, transmitted to a data-collecting station located on the sea shore. The ship or the data-collecting stations are central nodes. Denote H the set of all elements of an arbitrary UWSN. Suppose that all elements are capable of handling (i.e., receiving or transmitting) data in the same way. Also suppose that they perform the same set of tasks. Thus they are, from the mathematical point of view, interchangeable and equal (of course, with respect to their functionality as sinks and sensor nodes). The aim of the system is to collect information. Therefore, our elements of H must communicate data. This should be done ideally upwards, towards the surface. As we have mentioned above, there are different ways of passing information. In our model we concentrate on multipath and cluster routing approach (see Figure 1and Figure 2). For details concerning these see Ayaz et al. and Li et al. [ 4 , 29 ]. Multipath routing protocols (Figure 1), forward the data packets to the sink via other nodes while in cluster based routing protocols (Figure 2), data packets are first aggregated to the respective cluster heads and only then forwarded via other cluster heads to the sink. For our purposes, we denote the i –th cluster by cli . Its cluster head will be denoted by CHi. We call non-CH nodes ordinary and sinks will be treated as cluster heads. Now, suppose that the elements of our system are clustered. In other words, some elements of H function as cluster heads, i.e., masters, while others are ordinary. The data aggregation process goes as follows: Within their cluster, the ordinary elements pass information to their cluster head while between clusters, i.e., supposedly over longer distances, only cluster heads communicate. At a given point in time, each cluster has the unique cluster head, and each element can belong to exactly one cluster. We denote the i–th cluster by cliand its cluster head by CHi. Figure 1. Multipath approach to UWSN data aggregation. Notice the oriented communication between nodes. Symmetry 2019,11, 734 5 of 16 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. Now, for a given pair a , b∈H , regard a binary hyperoperation, where a∗b is, for arbitrary a,b∈H, defined by: a∗b=({a,b} ∪ [a·b)≤for (a=CHi,b=CHj)or a,b∈cli {a,b}for (a6=CHior b6=CHj)and (a∈cli,b∈clj,i6=j)(1) By [a·b)≤ we mean a set {x∈H|a·b≤x} , where a·b is a result of a single-valued binary operation such that a·bis, for arbitrary a,b∈H, defined by: a·b=     CHifor a,b∈cli CHkfor a=CHi,b=CHj,i6=j sfor ((a6=CHior b6=CHj)and (a∈cli,b∈clj,i6=j)) or a=sor b=s (2) and CHk is such a cluster head that CHi≤CHk , CHj≤CHk , where a≤b is a relation between elements of H such that: (1) s≤s , s≤CHi and CHi≤s for all clusters cli , (2) within the same cluster cli we have aj≤CHi for all aj∈cli while mutually different ordinary elements of the cluster are incomparable, (3) between clusters for a=CHi , b=CHj the fact that a≤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 CHk above we mean a cluster head on the closest tier above both CHi and CHj . Of course, CHk 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 CHi≤CHj and simultaneously CHj≤CHi does not mean that CHi=CHj , rather it only means that CHi and CHj 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 " ≤ " (restricted to H\ {s} ) becomes partial ordering and we can write CHk=sup{CHi , CHj} (with respect to the relation " ≤ "). 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 Symmetry 2019,11, 734 6 of 16 in fact write He=H∪ {s} , where He could mean "expanded". Of course, if we choose the option of s∈H, then He=H. Under these definitions, a·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·b=s or a∗b={a , b} or a∗b={a,b,s}all stand for communication failure. Lemma 1. [1](H,≤)is a quasi-ordered set. Suppose now that we have arbitrary a , b∈H . Since the result of a·b is such an element of H in which the data from a and b meet, it is natural to suppose that a·b=b·a , i.e., that " · " is commutative. However, we can suppose this only on condition that there exists such an algorithms that a·b=CHk=CHl=b·a for arbitrary clusters clk , cll . Further on suppose that such an algorithm exists, i.e., that (H,·)is a commutative groupoid. The following lemma is obvious. Lemma 2. [1]If (H,·)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)∩(a∗b)∗c6=∅for all a,b,c∈H; a quasi-hypergroup is a reproductive hypergroupoid. Lemma 3. [1]The hypergroupoid (H,∗)is a Hv–group, i.e., a weak associative quasi-hypergroup. Lemma 4. [ 1 ]The quasi-ordering " ≤ " and the operation " · " are compatible, i.e., for all a , b∈H such that a≤b and an arbitrary c ∈H there is a ·c≤b·c and c ·a≤c·b. Now, denote HCH ⊆H the set of cluster heads. This notation enables us to regard both clustering based systems and multipath systems because the fact that HCH =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. This is because there is no need for the special element s and we do not distinguish between communication within and between clusters. The operation " · " defined by Equation (2) reduces to a·b=c (we still suppose that it is commutative) and, consequently, the hyperoperation Equation (1) reduces to a∗b={a,b} ∪ [a·b)≤, in both cases for all a,b∈H. Lemma 5. [ 1 ]If we are able to uniquely identify CHk in Equation (2) , then (HCH , · , ≤) is a partially ordered semigroup. Finally, what is x∈[a)≤ ? This means that a≤x , i.e., that the data from the element a reach the element x . Thus, if x is a sink, than the fact that x∈[a)≤ means that the data from a can be successfully collected. What we want is that, if we denote S the set of all sinks, for all a∈H there exists at least one x∈S such that x∈[a)≤ , which means that data from all elements of our network H can be successfully collected. Of course, in order to achieve this, it is crucial to have an algorithm for unique determination of CHk in Equation (2) . Yet clustering algorithms such as the Distributed Underwater Clustering Scheme (DUCS) [ 3 ] or Low Energy Adaptive Clustering Hierarchy (LEACH) protocol can provide this. 3. Use of the Theory of Quasi-Automata In Definition 2, the concept of quasi-order hypergroup is defined. Chvalina and Chvalinová [ 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, Symmetry 2019,11, 734 7 of 16 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 Chvalina and Chvalinová [ 25 ] uses a general term automaton; however, notice that [ 25 ], p. 107, plain text, defines automaton in the way of Definition 4, which is a definition adopted by the authors of [ 25 ] in later years). For an overall discussion of the concepts and the reasons for our choice of the name see Novák et al. [30]. For some further reading and applications see also Hošková et al. [31–33]. Definition 4. By a quasi–automaton we mean a structure A= (I , S , δ) such that I6=∅ is a monoid, S6=∅ and δ:I×S→S satisfies the following condition: 1. There exists an element e ∈I such that δ(e,s) = s for any state s ∈S; 2. δ(y,δ(x,s)) = δ(xy,s)for any pair x,y∈I and any state s ∈S. The set I is called the input set or input alphabet, the set S is called the state set and the mapping δ is called next-state or transition function. Condition 2 is called GMAC (Generalized Mixed Associativity Condition). In [ 25 ], Chvalina and Chvalinová 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 , ◦) is a hypergroup, or rather quasi-order hypergroup, (hence the name state hypergroup) was proved in [25]. (Notice that in [25]Iand Sare swapped.) Definition 5. Let A= (I , S , δ) be an automaton. We define a binary hyperoperation " ◦ " on the state set S by: s◦t=δ(I∗,s)∪δ(I∗,t)(3) for any pair of states s , t∈S , where A∗ is a free monoid of words over the (non-empty) alphabet A . The hyperstructure (S,◦)is called state hypergroup of the automaton A. Some properties of automata following from properties of its state hypergroup are proved in [ 25 ]. This includes the properties of being connected or separated. Definition 6. Let A= (I , S , δ) be a quasi-automaton. A quasi-automaton B= (I , S1 , δ1) such that S1⊆S and δ1 is a restriction of δ on I×S1 and δ(a , s)∈S1 for any state s∈S1 and any word a∈I∗ , is called a sub quasi-automaton of A . A sub quasi-automaton B= (I , S1 , δ1) of a quasi-automaton A= (I , S , δ) is called separated if δ(S\S1 , I∗)∩S1=∅ . A quasi-automaton is called connected if it does not posses any separated proper subautomaton. A quasi-automaton A= (I , S , δ) is called strongly connected if for any states s , t∈S there exists a word a ∈I∗such that δ(a,s) = t. If in quasi–automata we suppose that the input set I is a semihypergroup instead of a free monoid, we arrive at the concept of a quasi–multiautomaton. When defining this concept, caution must be exercised when adjusting the conditions imposed on the transition function δ as on the left-hand side of condition 2 we get a state while on the right-hand side we get a set of states. However, in the dichotomy deterministic —nondeterministic, quasi–multiautomata still are deterministic because the range of δ is S . The difference between the transition function of a quasi-automaton and the transition function of a quasi-multiautomaton is that in quasi-automata the state achieved by applying y in a Symmetry 2019,11, 734 8 of 16 state, which is the result of application of x in s , is the same as the state achieved by applying xy in s , while condition (4) says that it is one of the many states achievable by applying any command from x∗yin state s. Definition 7. A quasi–multiautomaton is a triad A= (I , S , δ) , where (I , ∗) is a semihypergroup, S is a non–empty set, and δ:I×S→S is a transition map satisfying the condition: δ(b,δ(a,s)) ∈δ(a∗b,s)for all a,b∈I,s∈S. (4) The hyperstructure (I , ∗) is called the input semihypergroup of the quasi–multiautomaton A ( I alone is called the input set or input alphabet), the set S is called the state set of the quasi–multiautomaton A , and δ is called next-state or transition function. Elements of the set S are called states, elements of the set I are called input symbols. Further on, we will make use of the above mentioned concepts to model the process of data aggregation. Notice that in Novák et al. [ 30 ], Cartesian composition of automata resulting in a quasi-multiautomaton is used to describe a task from collective robotics. Moreover, in Chvalina et al. [18] , the issue of state sets and input sets having the form of vectors and matrices (of both numbers and special classes of functions) is discussed in the context of quasi-multiautomata. In Figure 2we can see that the elements of the UWSN are divided into several tiers. Also, the nodes are grouped into clusters. The process of data aggregation happens as follows: First, data is collected in cluster heads and then transmitted between cluster heads towards the surface, i.e., "upwards". Obviously, we can only transmit the amount of data that the capacity of available memory allows. Suppose that clusters cover areas of more or less the same size, i.e., it does not matter how many nodes there are in respective clusters. Now, regard a set of vectors: Sv={~ v= (v1,v2, . . . , vn)|vi∈ h0;1i;i∈ {1, . . . , n}} (5) of such a number of components that corresponds to the number of tiers (with index 1 meaning surface and index n meaning seabed or the deepest tier). The components vi carry information about how much total memory all cluster heads at a given tier has been used. In other words, v2= 0.6 means that at the second tier 60% memory capacity has been used, regardless of whether this 60% means that every cluster head at this level has 40% free capacity or whether 6 out of 10 cluster heads already have no available memory while 4 are 100% free. The process of data aggregation starts with ~ v= ( 0, . . . ,0 ) , i.e., at the moment when all cluster heads have empty memory. Since, at first data are collected within clusters, ~ v immediately becomes non-zero. The process of communication between cluster heads is described by the change of ~ v by means of multiplying~ vby a square matrix of real numbers: IM=     Ak=   a11 . . . a1n . . . . . . . . . an1. . . ann   |aij ∈ h0; 1ifor i≥j,aij =0 for i<j;i,j∈ {1, . . . , n};kAkk1≤1     , where kAkk1 is the column norm of Ak . In other words, IM is a subset of the set of upper triangular matrices, i.e., we can also write: IM=           Ak=      a11 0 0 . . . 0 a21 a22 0 . . . 0 . . .. . ..... . .. . . an1an2an3. . . ann       | kAkk1≤1           (6) Symmetry 2019,11, 734 9 of 16 Now, these two sets Svand IMwill be linked with a transition function δby: δ(A,~ v) = ~ v·A(7) for all A∈IM and all ~ v∈Sv . If we regard the usual matrix multiplication (only swapped), i.e., AB=B·A for all A , B∈IM , then (IM , ) is a monoid such that the free monoid I∗ M=IM . Thus we can regard the triple (IM , Sv , δ) 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 (IM,Sv,δ)is a quasi-automaton. Proof. The identity matrix En is the neutral element of IM . Property 1 of Definition 4holds trivially. Verification of Property 2 is also straightforward: δ(A,δ(B,~ v)) = δ(A,~ v·B) = (~ v·B)·A=~ v·B·A=δ(B·A,~ v) = δ(AB,~ v). The set IM consists of matrices such that the column norm ||A||1 is at most one. In the following Remark, we show that without this condition the set IMwould not be closed with respect to "". Remark 1. Suppose two matrices A,B∈IM. If we denote: A=      a11 0 0 . . . 0 a21 a22 0 . . . 0 . . .. . ..... . .. . . an1an2an3. . . ann       B=      b11 0 0 . . . 0 b21 b22 0 . . . 0 . . .. . ..... . .. . . bn1bn2bn3. . . bnn       , then, BA=A·B=            1 ∑ i=1a1ibi10 0 . . . 0 2 ∑ i=1a2ibi2 2 ∑ i=2a2ibi20 . . . 0 . . .. . ..... . .. . . n ∑ i=1anibin n ∑ i=2anibin n ∑ i=3anibin . . . n ∑ i=n anibin            and it is obvious that the column norm kBAk1 will not exceed 1, i.e., BA∈IM . Indeed, suppose that A is an all-ones matrix upper triangular matrix (which, of course violates the condition that kAk1≤ 1). Then all the sums in BA reduce to n ∑ i=j bij which are (due to the fact that kBk1≤ 1) smaller than 1. If moreover kAk1≤ 1, none of the sums becomes greater. Of course, in IM 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. Every tier has an arbitrary number of clusters, i.e., an arbitrary number of cluster heads, each with the same memory capacity. Then by vector e.g., ~ v= ( 0.3;0.1;0;0.5 ) , we describe such a state of the system that, at a certain Symmetry 2019,11, 734 16 of 16 30. Novák, M.; Kˇrehlík, Š.; Stanˇek, D. n–ary Cartesian composition of automata. Soft Comput. 2019. [CrossRef] 31. Hošková, Š. Discrete transformation hypergroups. 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