Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators

mathematics Article Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators Jan Chvalina 1, Michal Novák 1,* , Bedˇrich Smetana 2and David Stanˇek 1   Citation: Chvalina, J.; Novák, M.; Smetana, B.; Stanˇek, D. Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators. Mathematics 2021,9, 319. https://doi.org/10.3390/math9040319 Academic Editor: Christos G. Massouros Received: 29 December 2020 Accepted: 2 February 2021 Published: 5 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Mathematics, Faculty of Electrical Engineeering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic; [email protected].cz (J.C.); [email protected].cz (D.S.) 2Department of Quantitative Methods, University of Defence in Brno, Kounicova 65, 662 10 Brno, Czech Republic; [email protected] *Correspondence: [email protected].cz; Tel.: +420-541146077 Abstract: The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups. Keywords: hyperstructure theory; linear differential operators; ODE; automata theory 1. Introduction This paper discusses sequences of groups, hypergroups and automata of linear differential operators. It is based on the algebraic approach to the study of linear ordinary differential equations. Its roots lie in the work of Otakar Bor˚uvka, a Czech mathematician, who tied the algebraic, geometrical and topological approaches, and his successor, František Neuman, who advocated the algebraic approach in his book [ 1 ]. Both of them (and their students) used the classical group theory in their considerations. In several papers, published mainly as conference proceedings such as [ 2 – 4 ], the existing theory was extended by the use of hypercompositional structures in place of the usual algebraic structures. The use of hypercompositional generalizations has been tested in the automata theory, where it has brought several interesting results; see, e.g., [ 5 – 8 ]. Naturally, this approach is not the only possible one. For another possible approach, investigations of differential operators by means of orthognal polynomials, see, e.g., [9,10]. Therefore, in this present paper we continue in the direction of [ 2 , 4 ] presenting results parallel to [ 11 ]. Our constructions, no matter how theoretical they may seem, are motivated by various practical issues of signal processing [ 12 – 16 ]. We construct sequences of groups and hypergroups of linear differential operators. This is because, in signal processing (but also in other real-life contexts), two or more connecting systems create a standing higher system, characteristics of which can be determined using characteristics of the original systems. Cascade (serial) and parallel connecting of systems of signal transfers are used in this. Moreover, series of groups motivated by the Galois theory of solvability of algebraic equations and the modern theory of extensions of fields, are often discussed in literature. Notice also paper [ 11 ] where the theory of artificial neurons, used further on in some examples, has been studied. Another motivation for the study of sequences of hypergroups and their homomorphisms can be traced to ideas of classical homological algebra which comes from the Mathematics 2021,9, 319. https://doi.org/10.3390/math9040319 https://www.mdpi.com/journal/mathematics Mathematics 2021,9, 319 2 of 16 algebraic description of topological spaces. A homological algebra assigns to any topological space a family of abelian groups and to any continuous mapping of topological spaces a family of group homomorphisms. This allows us to express properties of spaces and their mappings (morphisms) by means of properties of the groups or modules or their homomorphisms. Notice that a substantial part of homology theory is devoted to the study of exact short and long sequences of the above mentiones structures. 2. Sequences of Groups and Hypergroups: Definitions and Theorems 2.1. Notation and Preliminaries It is crucial that one understands the notation used in this paper. Recall that we study, by means of algebra, linear ordinary differential equations. Therefore, our notation, which follows the original model of Bor˚uvka and Neuman [ 1 ], uses a mix of algebraic and functional notation. First, we denote intervals by J and regard open intervals (bounded or unbounded). Systems of functions with continuous derivatives of order k on J are denoted by Ck(J) ; for k= 0 we write C(J) instead of C0(J) . We treat Ck(J) as a ring with respect to the usual addition and multiplication of functions. We denote by δij the Kronecker delta, i , j∈N , i.e., δii =δjj = 1 and δij = 0, whenever i6=j ; by δij we mean 1 −δij . Since we will be using some notions from the theory of hypercompositional structures, recall that by P(X) one means the power set of Xwhile (P)∗(X)means P(X)\∅. We regard linear homogeneous differential equations of order n≥ 2 with coefficients, which are real and continuous on J , and–for convenience reasons–such that p0(x)> 0 for all x∈J, i.e., equations y(n)(x) + pn−1(x)y(n−1)(x) + · · · +p0(x)y(x) = 0. (1) By Anwe, adopting the notation of Neuman [1], mean the set of all such equations. Example 1. The above notation can be explained on an example taken from [ 17 ], in which Neuman considers the third-order linear homogeneous differential equation y000(x)−q0 1(x) q1(−x)y00(x) + (q1(x)−1)2y0(x)−q0 1(x) q1(x)y(x) = 0 on the open interval J ∈R. One obtains this equation from the system y0 1=y2 y0 2=−y1+q1(x)y3 y0 3=−q1(x)y2 Here q1∈C+(J) satisfies the condition q1(x)6= 0on J . In the above differential equation we have n= 3, p0(x) = −q0 1(x) q1(x) , p1(x) = (q1(x)− 1 )2 and p2(x) = −q0 1(x) q1(−x) . It is to be noted that the above three equations form what is known as set of global canonical forms for the third-order equation on the interval J. Denote Ln(pn−1 , . . . , p0):Cn(J)→Cn(J) the above linear differential operator defined by Ln(pn−1, . . . , p0)y(x) = y(n)(x) + n−1 ∑ k=0 pk(x)y(k)(x), (2) where y(x)∈Cn(J) and p0(x)> 0 for all x∈J . Further, denote by LAn(J) the set of all such operators, i.e., LAn(J) = {L(pn−1, . . . , p0)|pk(x)∈C(J),p0(x)>0}. (3) Mathematics 2021,9, 319 3 of 16 By LAn(J)m we mean subsets of LAn(J) such that pm∈C+(J) , i.e., there is pm(x)> 0 for all x∈J . If we want to explicitly emphasize the variable, we write y(x) , pk(x) , etc. However, if there is no specific need to do this, we write y , pk , etc. Using vector notation ~ p(x) = (pn−1(x), . . . , p0(x)), we can write Ln(~ p)y=y(n)+ n−1 ∑ k=0 pky(k). (4) Writing L(~ p)∈LAn(J) (or L(~ p)∈LAn(J)m ) is a shortcut for writing Ln(~ p)y∈LAn(J) (or, Ln(~ p)y∈LAn(J)m). On the sets of linear differential operators, i.e., on sets LAn(J) , or their subsets LAn(J)m , we define some binary operations, hyperoperations or binary relations. This is possible because our considerations happen within a ring (of functions). For an arbitrary pair of operators L(~ p) , L(~ q)∈LAn(J)m , where ~ p= (pn−1 , . . . , p0) , ~ q= (qn−1 , . . . , q0) , we define an operation “ ◦m ” with respect to the m -th component by L(~ p)◦mL(~ q) = L(~ u), where ~ u= (un−1, . . . , u0)and uk(x) = pm(x)qk(x) + (1−δkm)pk(x)(5) for all k=n− 1, . . . ,0, k6=m and all x∈J . Obviously, such an operation is not commutative. Moreover, apart from the above binary operation we can define also a relation “ ≤m ” comparing the operators by their m -th component, putting L(~ p)≤mL(~ q) whenever, for all x∈J, there is pm(x) = qm(x)and at the same time pk(x)≤qk(x)(6) for all k=n−1, . . . , 0. Obviously, (LAn(J)m,≤m)is a partially ordered set. At this stage, in order to simplify the notation, we write LAn(J) instead of LAn(J)m because the lower index m is kept in the operation and relation. The following lemma is proved in [2]. Lemma 1. Triads (LAn(J),◦m,≤m)are partially ordered (noncommutative) groups. Now we can use Lemma 1to construct a (noncommutative) hypergroup. In order to do this, we will need the following lemma, known as Ends lemma; for details see, e.g., [18–20] . Notice that a join space is a special case of a hypergroup–in this paper we speak of hypergroups because we want to stress the parallel with groups. Lemma 2. Let (H , · , ≤) be a partially ordered semigroup. Then (H , ∗) , where ∗:H×H→ H is defined, for all a,b∈H by a∗b= [a·b)≤={x∈H|a·b≤x}, is a semihypergroup, which is commutative if and only if “ · ” is commutative. Moreover, if (H , ·) is a group, then (H,∗)is a hypergroup. Thus, to be more precise, defining ?m:LAn(J)×LAn(J)→ P(LAn(J)), (7) by L(~ p)?mL(~ q) = {L(~ u)|L(~ p)◦mL(~ q)≤mL(~ u)}(8) for all pairs L(~ p),L(~ q)∈LAn(J)m, lets us state the following lemma. Lemma 3. Triads (LAn(J),?m)are (noncommutative) hypergroups. Mathematics 2021,9, 319 4 of 16 Notation 1. Hypergroups (LAn(J),?m)will be denoted by HLAn(J)mfor an easier distinction. Remark 1. As a parallel to (2)and (3)we define L(qn, . . . , q0)y(x) = n ∑ k=0 qk(x)y(k)(x),q06=0, qk∈C(J)(9) and LAn(J) = {qn, . . . , q0)|q06=0, qk(x)∈C(J)}(10) and, by defining the binary operation “ ◦m ” and “ ≤m ” in the same way as for LAn(J)m , it is easy to verify that also (LAn(J) , ◦m , ≤m) are noncommutative partially ordered groups. Moreover, given a hyperoperation defined in a way parallel to (8) , we obtain hypergroups (LAn(J)m , ?m) , which will be, in line with Notation 1, denoted HLAn(J)m. 2.2. Results In this subsection we will construct certain mappings between groups or hypergroups of linear differential operators of various orders. The result will have a form of sequences of groups or hypergroups. Define mappings Fn:LAn(J)→LAn−1(J)by Fn(L(pn−1, . . . , p0)) = L(pn−2, . . . , p0) and φn:LAn(J)→LAn−1(J)by φn:(L(pn−1, . . . , p0)) = L(pn−2, . . . , p0). It can be easily verify that both Fn and φn are, for an arbitrary n≥ 2, group homomorphisms. Evidently, LAn(J)⊂LAn(J) , LAn−1(J)⊂LAn(J) for all admissible n∈N . Thus we obtain two complete sequences of ordinary linear differential operators with linking homomorphisms Fnand φn: LA0(J)id0,1 //LA1(J)id1,2 //LA2(J)id2,3 //. . . LA0(J) id0 OO LA1(J) id1 OO F1 oo φ1 ii LA2(J) id2 OO F2 oo φ2 hh . . . F3 oo φ3 gg . . . LAn−2(J)idn−2,n−1//LAn−1(J)idn−1,n//LAn(J)idn,n+1//. . . . . . LAn−2(J) idn−2 OO LAn−1(J) idn−1 OO Fn−1 oo φn−2 ii LAn(J) idn OO Fn oo φn hh . . . Fn+1 oo φn+1 gg (11) where idk,k+1,idkare corresponding inclusion embeddings. Notice that this diagram, presented at the level of groups, can be lifted to the level of hypergroups. In order to do this, one can use Lemma 3and Remark 1. However, this is not enough. Yet, as Lemma 4suggests, it is possible to show that the below presented assignment is functorial, i.e., not only objects are mapped onto objects but also morphisms (isotone group homomorphisms) are mapped onto morphisms (hypergroup homomorphisms). Notice that Lemma 4was originally proved in [ 4 ]. However, given the minimal impact of the proceedings and its very limited availability and accessibility, we include it here with a complete proof. Mathematics 2021,9, 319 5 of 16 Lemma 4. Let (Gk , ·k , ≤k) , k= 1,2 be preordered groups and f:(G1 , ·1 , ≤1)→(G2 , ·2 , ≤2) a group homomorphism, which is isotone, i.e., the mapping f:(G1 , ≤1)→(G2 , ≤2) is orderpreserving. Let (Hk , ∗k) , k= 1,2 be hypergroups constructed from (Gk , ·k , ≤k) , k= 1,2 by Lemma 2 , respectively. Then f:(H1 , ∗1)→(H2 , ∗2) is a homomorphism, i.e., f(a∗1b)⊆ f(a)∗2f(b)for any pair of elements a,b∈H1. Proof. Let a , b∈H1 be a pair of elements and c∈f(a∗1b) be an arbitrary element. Then there is d∈a∗1b= [a·1b)≤1 , i.e., a·1b≤1d such that c=f(d) . Since the mapping f is an isotone homomorphism, we have f(a)·2f(b) = f(a·1b)≤f(d) = c , thus c∈ [f(a)·2f(b))≤2. Hence f(a∗1b) = f([a·1b)≤1)⊆[f(a)·2f(b))≤=f(a)∗2f(b). Consider a sequence of partially ordered groups of linear differential operators LA0(J)F1 ←− LA1(J)F2 ←− LA2(J)F3 ←− . . . . . . Fn−2 ←−− LAn−2(J)Fn−1 ←−− LAn−1(J)Fn ←− LAn(J)Fn+1 ←−− LAn+1(J)←. . . given above with their linking group homomorphisms Fk:LAk(J)→LAk−1(J) for k=1, 2, . . . . Since mappings Fn:LAn(J)→LAn−1(J), or rather Fn:(LAn(J),◦m,≤m)→(LAn−1(J),◦m,≤m), for all n≥ 2, are group homomorphisms and obviously mappings isotone with respect to the corresponding orderings, we immediately get the following theorem. Theorem 1. Suppose J⊆R is an open interval, n∈N is an integer n= 2, m∈N such that m5n .Let (HLAn(J)m , ∗m) be the hypergroup obtained from the group (LAn(J)m , ◦m) by Lemma 2. Suppose that Fn:(LAn(J)m , ◦m)→(LAn−1(J)m , ◦m) are the above defined surjective group-homomorphisms, n∈N , n= 2. Then Fn:(HLAn(J)m , ∗m)→HLAn−1(J)m , ∗m) are surjective homomorphisms of hypergroups. Proof. See the reasoning preceding the theorem. Remark 2. It is easy to see that the second sequence from (11) can be mapped onto the sequence of hypergroups HLA0(J)m F1 ←− HLA1(J)m F2 ←− HLA2(J)m F3 ←− . . . . . . Fn−2 ←−− HLAn−1(J)m Fn−1 ←−− HLAn(J)m←. . . This mapping is bijective and the linking mappings are surjective homomorphisms Fn . Thus this mapping is functorial. 3. Automata and Related Concepts 3.1. Notation and Preliminaries The concept of an automaton is mathematical interpretation of diverse real-life systems that work on a discrete time-scale. Various types of automata, called also machines, are applied and used in numerous forms such as money changing devices, various calculating machines, computers, telephone switch boards, selectors or lift switchings and other technical objects. All the above mentioned devices have one aspect in common–states are switched from one to another based on outside influences (such as electrical or mechanical impulses), called inputs. Using the binary operation of concatenation of chains of input Mathematics 2021,9, 319 6 of 16 symbols one obtains automata with input alphabets in the form of semigroups or a groups. In the case of our paper we work with input sets in the form of hypercompositional structures. When focusing on the structure given by transition function and simultaneously neglecting the output functions and output sets, one reaches a generalization of automata– quasi-automata (or semiautomata); see classical works such as, e.g., [3,18,21–24]. To be more precise, a quasi-automaton is a system (A , S , δ) which consists of a nonvoid set A, an arbitrary semigroup Sand a mapping δ:A×S→Asuch that δ(δ(a,r,s)) = δ(a,r,s)(12) for arbitrary a∈A and r , s∈S . Notice that the concept of quasi-automaton has been introduced by S. Ginsberg as quasi-machine and was meant to be a generalization of the Mealy-type automaton. Condition (12) is sometimes called Mixed Associativity Condition (MAC). With most authors it is nameless, though. For further reading on automata theory and its links to the theory of hypercompositional structures (also known as algebraic hyperstructures), see, e.g., [ 24 – 26 ]. Furthermore, for clarification and evolution of terminology, see [ 8 ]. For results obtained by means of quasi-multiautomata, see, e.g., [5–8,27]. Definition 1. Let A be a nonempty set, (H , ·) be a semihypergroup and δ:A×H→A a mapping satisfying the condition δ(δ(s,a),b)∈δ(s,a·b)(13) for any triad (s , a , b)∈A×H×H , where δ(s , a·b) = {δ(s , x) ; x∈a·b} . Then the triad (A , H , δ) is called quasi-multiautomaton with the state set A and the input semihypergroups (H , ·) . The mapping δ:A×H→A is called the transition function (or the next-state function) of the quasi-multiautomaton (A , H , δ) . Condition (13) is called Generalized Mixed Associativity Condition (or GMAC). In this section, Rn[x]means, as usually, the ring of polynomials of degree at most n. 3.2. Results Now, consider linear differential operators L(m , pn−1 , . . . , p0):C∞(R)→C∞(R) defined by L(m,pn−1, . . . , p0)f=mdnf(x) dxn+ n−1 ∑ k=0 pk(x)dkf(x) dxk. (14) Denote by LA1An(R) the additive abelian group of differential operators L(m , pn−1 , . . . , p0), where for L(m,pn−1, . . . , p0),L(k,qn−1, . . . , q0)∈LA1An(R)we define L(m,pn−1, . . . , p0) + L(k,qn−1, . . . , q0) = L(m+k,pn−1+qn−1, . . . , p0+q0), (15) where L(m+k,pn−1+qn−1, . . . , p0+q0)f= (m+k)dnf(x) dxn+ n−1 ∑ k=0 (pk(x) + qk(x))dkf(x) dxk. (16) Suppose that pk∈Rn−1[x]and define δn: Rn[x]×LA1An(R)→Rn[x](17) by δn(f,L(m,pn−1, . . . , p0)) = mdnf(x) dxn+f(x) + m+ n−1 ∑ k=0 pk(x),f∈Rn[x]. (18) Mathematics 2021,9, 319 7 of 16 Theorem 2. Let LA1An(R) , Rn[x] be structures and δn:Rn[x]×LA1An(R)→Rn[x] the mapping defined above. Then the triad (Rn[x] , LA1An(R) , δn) is a quasi-automaton, i.e., an action of the group LA1An(R)on the group Rn[x]. Proof. We are going to verify the mixed associativity condition (MAC) which should satisfy the above defined action: Suppose f∈Rn[x] , f(x) = ∑n k=0akxk , L(m , pn−1 , . . . , p0) , L(k , qn−1 , . . . , q0) ∈LA1An(R). Then δn(δn(f,L(m,pn−1, . . . , p0)),L(k,qn−1, . . . , q0)) = =δn mdnf(x) dxn+f(x) + m+ n−1 ∑ k=0 pk(x),L(k,qn−1, . . . , q0)!= =δn m·n!·an+m+f(x) + n−1 ∑ k=0 pk(x),L(k,qn−1, . . . , q0)!= =kdnf(x) dxn+m·n!·an+m+f(x) + n−1 ∑ k=0 pk(x) + n−1 ∑ k=0 qk(x) + k= = (m+k)n!·an+ (m+k) + f(x) + n−1 ∑ k=0 (pk(x) + qk(x))= = (m+k)(n!·an+1) + f(x) + n−1 ∑ k=0 (pk(x) + qk(x))= = (m+k)dnf(x) dxn+f(x) + (m+k) + n−1 ∑ k=0 (pk(x) + qk(x))= =δn(f,L(m+k,pn−1+qn−1, . . . , p0+q0))= =δn(f,L(m,pn−1, . . . , p0) + L(k,qn−1, . . . , q0)), (19) thus the mixed associativity condition is satisfied. Since Rn[x] , LA1An(R) are endowed with naturally defined orderings, Lemma 2can be straightforwardly applied to construct semihypergroups from them. Indeed, for a pair of polynomials f , g∈Rn[x] we put f≤g , whenever f(x)≤ g(x) , z∈Rn[x] . In such a case (Rn[x],≤) is a partially ordered abelian group. Now we define a binary hyperoperation # : Rn[x]×Rn[x]→ P?(Rn[x])(20) by f#g={h;h∈Rn[x],f(x) + g(x)≤h(x),x∈R}= [f+g)≤. (21) By Lemma 2we have that (Rn[x], #)is a hypergroup. Moreover, defining # : LA1An(R)×LA1An(R)→ P?(LA1An(R)) (22) by L(m , −−→ p(x)) # L(k , −−→ q(x)) = hL(m,−−→ p(x)) + L(k,−−→ q(x))≤=hL(m+k,−−→ p(x) + −−→ q(x)≤= {L(r,−−→ u(x));m+k≤r,−−→ p(x) + −−→ q(x)≤−−→ u(x)}, which means pj(x) + qj(x)≤uj(x), where j= 0,1, . . . , n− 1, we obtain, again by Lemma 2that the hypergroupoid (LA1An(R),#) is a commutative semihypergroup. Mathematics 2021,9, 319 8 of 16 Finally, define a mapping σn:LA1An(R)×Rn[x]→Rn[x](23) by σn(L(m,pn−1, . . . , p0,f)) = L(m,p0◦f+pn−1, . . . , p0◦f+p1,p0). (24) Below, in the proof of Theorem 3, we show that the mapping satisfies the GMAC condition. This allows us to construct a quasi-multiautomaton. Theorem 3. Suppose (LA1An(R) ,# ) , (Rn[x] ,# ) are hypergroups constructed above and σn: LA1An(R)×Rn[x]→Rn[x]is the above defined mapping. Then the structure ((LA1An(R),#),(Rn[x],#),σn) is a quasi-multiautomaton. Proof. Suppose L(m,~ p)∈LA1An(R),f,g∈Rn[x]. Then σn(σn(L(m,~ p),f),g)=σn(L(m,p◦f+pn−1, . . . , p◦f+p1,p0),g) = =L(m,p◦g+p◦f+pn−1, . . . , p◦g+p◦f+p1,p0) = =L(m,p◦(g+f) + pn−1, . . . p◦(g+f) + p1,p0)∈ ∈ {σn(L(m,p◦h+pn−1, . . . p◦h+p1,p0);f,g,h∈Rn[x],f+g≤h}= =σn(L,m,pn−1, . . . , p1,p0),[f+g)≤) = σn(L(m,~ p),f#g), (25) hence the GMAC condition is satisfied. Now let us discuss actions on objects of different dimensions. Recall that a homomorphism of automaton (S , G , δS) into the automaton (T , H , δT) is a mapping F=φ×ψ: S×G→T×H such that φ:S→T is a mapping and ψ:G→H is a homomorphism (of semigroups or groups) such that for any pair [s,g]∈S×Gwe have φ(δS(s,g)) = δT(φ(s),ψ(g)), i.e., φ◦δS=δT◦(φ×ψ). (26) In order to define homomorphisms of our considered actions and especially in order to construct a sequence of quasi-automata with decreasing dimensions of the corresponding objects, we need a different construction of a quasiautomaton. If f∈Rn[x],f(x) = ∑n k=0anxkand L(m,~ p)∈LA1An(R), we define τn(L(m,pn−1, . . . , p0),f) = L(m,an+pn−1, . . . , a1+p0+a0). (27) Now, if g∈Rn[x],g(x) = ∑n k=0bkxk, we have τn(τn(L(m,pn−1, . . . , p0),f),g) = =τn(L(m,an+pn−1, . . . , a1+p1+a0),g) = =L(m,an+bn+pn−1, . . . , a1+b1+p0+a0+b0) = =τn L(m,pn−1, . . . , p0),n ∑ k=0 (ak+bk)x!= =τn(L(m,pn−1, . . . , p0),f+g). (28) Hence τn:LA1An(R)×Rn[x]→LA1An(R) is the transition function (satisfying MAC) of the automaton A=(LA1An(R),Rn−1[x],τn). Mathematics 2021,9, 319 9 of 16 Consider now two automata– An−1=(LA1An−1(R),Rn−1[x],τn−1) and the above one. Define mappings φn:LA1An(R)→LA1An−1(R),ψn:Rn[x]→Rn−1[x](29) in the following way: For L(m,pn−1, . . . , p0)∈LA1An(R)put φn(L(m,pn−1, . . . , p0))=L(m,pn−2, . . . , p0)∈LA1An−1(R)(30) and for f∈Rn[x],f(x) = ∑n k=0akxkdefine ψn(f) = ψn n ∑ k=0 akxk!= n−1 ∑ k=0 akxk∈Rn−1[x]. (31) Evidently, there is ψn(f+g) = ψn(f) + ψ(g) for any pair of polynomials f , g∈Rn[x] . Theorem 4. Let φn:LA1An(R)→LA1An−1(R) , ψn:Rn[x]→Rn−1[x] , τn:LA1An(R)× Rn[x]→LA1An(R) , n∈N , n= 2, be mappings defined above. Define Fn:An→ An−1 as mapping Fn=φn×ψn:LA1An(R)×Rn[x]→LA1An−1(R)×Rn−1[x]. Then the following diagram LA1An(R)×Rn[x]τn// φn×ψn  LA1An(R) φn  LA1An−1(R)×Rn−1[x]τn−1//LA1An−1(R) (32) is commutative, thus the mapping Fn=φn×ψn is a homomorphism of the automaton An= (LA1An(R),Rn[x],τn)into the automaton An−1= (LA1An−1(R),Rn−1[x],τn−1). Proof. Let [L(m,~ p),f]∈LA1An(R)×Rn[z],f(x) = ∑n k=0akxk. Then (φn◦τn)(L(m,~ p),f)=φn τn L(m,pn−1, . . . , p0),n ∑ k=0 akxk!!= =φn((m,an+pn−1, . . . , a1+p0+a0))=L(m,an−1+pn−2, . . . , a1+p0+a0) = =τn−1 L(m,pn−2, . . . , p0),n−1 ∑ k=0 akxk!= =τn−1 (φn×ψn) L(m,pn−1, . . . , p0),n ∑ k=0 akxk!!= =(τn−1◦(φn×φn)))(L(m,~ p),f),(33) Thus the diagram (32) is commutative. Using the above defined homomorphism of automata we obtain the sequence of automata with linking homomorphisms Fk:Ak→ An−1,k∈N,k≥2 : . . . Fn−1 ←−− (LA1An−1(R),Rn−1[x],τn−1)Fn ←− (LA1An(R),Rn[x],τn) (LA1A1(R),τ1)F2 ←− (LA1A2(R),τ2)F3 ←− . . . Fn−2 ←−− (LA1An−2(R),Rn−2[x],τn−2)Fn−1 ←−− . . . (34) Mathematics 2021,9, 319 16 of 16 18. Kˇrehlík, Š.; Novák, M. From lattices to Hv–matrices. An. ¸St. Univ. Ovidius Constan¸ta 2016,24, 209–222. [CrossRef] 19. Novák, M. On EL-semihypergroups. Eur. J. Comb. 2015,44, 274–286; ISSN 0195-6698. [CrossRef] 20. Novák, M. Some basic properties of EL-hyperstructures. Eur. J. Comb. 2013,34, 446–459. [CrossRef] 21. Bavel, Z. 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