n-ary hyperstructures constructed from binary quasi-ordered semigroups

DOI: 10.2478/auom-2014-0056 An. S¸t. Univ. Ovidius Constant¸a Vol. 22(3),2014,147–168 n–ary hyperstructures constructed from binary quasi–ordered semigroups Michal Nov´ak Abstract Based on works by Davvaz, Vougiouklis and Leoreanu-Fotea in the field of n–ary hyperstructures and binary relations we present a construction of n–ary hyperstructures from binary quasi-ordered semigroups. We not only construct the hyperstructures but also study their important elements such as identities, scalar identities or zeros. We also relate the results to earlier results obtained for a similar binary construction and include an application of the results on a hyperstructure of linear differential operators. 1 Introduction Since its introduction in 1930s, the study of binary hyperstructures has become an established area of research thanks to authors of numerous papers on the topic as well as thanks to standard books which sum up the basic concepts of hyperstructure theory and their applications. Yet the step from binary hyperstructures to n–ary hyperstructures has been done only recently by Davvaz and Vougiouklis who in [13] introduced the concept of n–ary hypergroup (sometimes called simply n–hypergroup) and presented n–ary generalization of some very basic concepts of hyperstructure theory. Apart from [13] the origins of our paper can be traced back to the issue introduced to hyperstructure theory by Rosenberg, Corsini, Leoreanu-Fotea, Key Words: hyperstructures, n–ary hyperstructures, partially ordered and quasi-ordered sets.2010 Mathematics Subject Classification: Primary 20N20, 06F15; Secondary 06F05. Received: 25 July, 2013. Revised: 1 October, 2013. Accepted: 8 October, 2013. 147 N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 148 Chvalina and others in works such as [3, 4, 10, 11, 23], i.e. the relation of hyperstructures and binary relations. Some particular constructions of hyperstructures associated to quasi-ordered single-valued structures introduced by Chvalina in [3, 4] have been studied and developped by Corsini, Davvaz, Heidari, Hoˇskov´a–Mayerov´a, Nezhad, and others in works such as [5, 8, 10, 14, 21]. This paper generalizes one of Chvalina’s constructions of binary hyperstructures from single-valued quasi-ordered semigroups. Results recently obtained in the area of n–ary generalization of hyperstructures associated to binary relations fall into three groups: some, such as Cristea and S¸tef˘anescu in e.g. [7, 9], generalize the binary relation and construct binary hyperstructures associated to n–ary relations while others, such as Leoreanu-Fotea and Davvaz in e.g. [17] generalize the hyperstructure and construct n–ary hyperstructures associated to binary relations. Finally, the third approach, presented e.g. in [1] is possible too – as one can study n–ary hyperstructures associated to n–ary relations. Out of these three options we develop the approach pioneered by Leoreanu-Fotea and Davvaz in [17]. We make use of n–ary hyperstructure concepts defined in [2, 13, 15]. As far as the basic binary concepts of hyperstructure theory are concerned, we use their definitions and meaning included in [10, 12]. For respective definitions see section 2 or respective places in the paper. Sometimes the definitions are adjusted in order to keep unified form of notation and/or naming throughout the paper. This is especially true for definitions and theorems taken from [2]. Notice that the original contruction, which is generalized in this paper, can be used in a number of contexts including differential equations, integral and integro–differential equations (hyperstructures of linear differential operators, Fredholm and Volterra equations), microeconomics (preference relations), chemistry, genetics, etc. For details cf. references of papers written on the topic by authors such as Chvalina, Hoˇskov´a–Mayerov´a, Raˇckov´a or Nov´ak. Some more examples may be found in [21] and its references. Finally, notice that the study of n–ary hyperstructures has important implications in the study of fuzzy hyperstructures and that the connection between hypergroups and n–ary hypergroups has been thoroughly studied in [16]. 2 Basic notions and concepts In the paper we work with the generalization of the basic concepts of the hyperstructure theory such as (binary) hyperoperation,semihypergroup and hypergroup. For their definitions cf. e.g. [10, 12]. Further, we work with the following three definitions included in [13] in the following wording: Definition 2.1. Let Hbe a non-empty set and fbe a mapping f:H×H→ P∗(H), where P∗(H)is the set of all non-empty subsets of H. Then fis N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 149 called a binary hyperoperation of H. We denote by Hnthe cartesian product H×. . . ×H, where Happears ntimes. An element of Hnwill be denoted by (x1, . . . , xn), where xi∈Hfor any iwith 1≤i≤n. In general, a mapping f:Hn→P∗(H)is called an n–ary hyperoperation and nis called the arity of hyperoperation. Let fbe an n–ary hyperoperation on Hand A1, . . . , An subsets of H. We define f(A1, . . . , An) = ∪{f(x1, . . . , xn)|xi∈Ai, i = 1, . . . , n}. We shall use the following abbreviated notation: the sequence xi, xi+1, . . . , xj will be denoted by xj i. For j < i, xj iis the empty set. In this convention f(x1, . . . , xi, yi+1, . . . , yj, zj+1, . . . , zn) will be written as f(xi 1, yj i+1, zn j+1). Definition 2.2. A non-empty set Hwith an n–ary hyperoperation f:Hn→ P∗(H)will be called an n–ary hypergroupoid and will be denoted by (H, f). An n–ary hypergroupoid (H, f)will be called an n–ary semihypergroupoid if and only if the following associative axiom holds: f(xi−1 1, f(xn+i−1 i), x2n−1 n+i) = f(xj−1 1, f(xn+j−1 j), x2n−1 n+j) (1) for every i, j ∈ {1,2, . . . , n}and x1, x2, . . . , x2n−1∈H. Definition 2.3. An n–ary semihypergroup (H, f)in which the equation b∈f(ai−1 1, xi, an i+1) (2) has the solution xi∈Hfor every a1, . . . , ai−1, ai+1, . . . , an, b ∈Hand 1≤i≤ n, is called an n–ary hypergroup. Notice that [17] uses the names n–semihypergroup and n–hypergroup instead. With respect to Definition 2.3 also notice that in our paper, especially in Theorem 4.3, we make use of an equivalent definition of the hypergroup by means of generalization of the reproductive axiom. For details cf. p. 156 or [13], p. 167. In the paper we also use generalizations of the concept of identity,scalar identity,zero element and inverse. The respective n–ary definitions are included in section 5 of the paper. Notice that in the binary context we use them in the following meaning. Definition 2.4. An element e∈H, where (H, ∗)is a hyperstructure, is called an identity if for all x∈Hthere holds x∗e3x∈e∗x. If for all x∈Hthere N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 150 holds x∗e={x}=e∗x, then e∈His called a scalar identity. If (H, ∗)is a hypergroup endowed with at least one identity, then an element a0∈His called an inverse of a∈Hif there is an identity e∈Hsuch that a∗a03e∈a0∗a. An element 0∈His called a zero element of Hif for all x∈Hthere holds x∗0 = {0}= 0 ∗x. Notice that the zero element of Definition 2.4 is sometimes called absorbing element or zero scalar element or simply zero scalar. Study of elements with the above properties (usually when combined in hyperstructures with two (hyper)operations) is important especially in the context of various types of ring-like hyperstructures or hyperideals. For implications in the area of (binary) EL–hyperstructures cf. [20], for some implications in the theory of hyperideals (in n–ary context) cf. e.g. [2]. 3 The binary construction and nature of its n–ary extension The original construction, which we are going to extend, has first been presented in [4] in the following form. Lemma 3.1. ([4], Theorem 1.3, p. 146) Let (S, ·,≤)be a partially ordered semigroup. Binary hyperoperation ∗:S×S→P0(S)defined by a∗b= [a·b)≤(3) is associative. The semi-hypergroup (S, ∗)is commutative if and only if the semigroup (S, ·)is commutative. 2 The hyperstructure (S, ∗) constructed in this way is usually called the associated hyperstructure to the single-valued structure (S, ·) or an ”Ends lemma”–based hyperstructure, or an EL–hyperstructure for short. The carrier set is denoted by Sif it is a semigroup or Hif it is a group. Lemma 3.2. ([4], Theorem 1.4, p. 147) Let (S, ·,≤)be a partially ordered semigroup. The following conditions are equivalent: 10For any pair (a, b)∈S2there exists a pair (c, c0)∈S2such that b·c≤a and c0·b≤a 20The associated semi-hypergroup (S, ∗)is a hypergroup. 2 N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 151 Remark 3.3. If (S, ·,≤)is a partially ordered group, then if we take c=b−1·a and c0=a·b−1, then condition 10is valid. Therefore, if (S, ·,≤)is a partially ordered group, then its associated hyperstructure is a hypergroup. Remark 3.4. The wording of the above lemmas is the exact translation of lemmas from [4]. The respective proofs, however, do not change in any way, if we regard quasi-ordered structures instead of partially ordered ones as the anti-symmetry of the relation ≤is not needed (with the exception of the ⇐implication of the part on commutativity, which does not hold in this case). The often quoted version of the ”Ends lemma” is therefore the version assuming quasi–ordered structures. Example 3.5. Regard the set (R,+,≤), i.e. the partially ordered group of real numbers. Obviously, (R, ∗), where a∗b= [a+b)≤={x∈R;a+b≤x} for arbitrary real numbers a, b, is a commutative hypergroup. Example 3.6. Regard the set (P∗(S),∪,⊆)of all non-empty subsets of an arbitrary set S. Obviously, (P∗(S),∪,⊆)is a partially ordered semigroup which is not a group and (P∗(S),∗), where A∗B= [A∪B)⊆={X∈P∗(S); A∪B⊆X} for arbitrary subsets A, B of S, is a commutative semihypergroup. One can prove that it is not a hypergroup. However, one can prove that by including ∅ we get a hypergroup. In other words, EL–hyperstructures are hyperstructures of arity 2. It is thus natural to find out whether the construction can be extended to involve more than two elements. Analogically to (3) we could define an n-ary hyperoperation ∗:S×. . . ×S | {z } n →P∗(S) by a1∗. . . ∗an | {z } n = [a1·. . . ·an | {z } n )≤={x∈S;a1·. . . ·an | {z } n ≤x}(4) In a standard notation used e.g. by [13] or [17] this would be denoted as a hyperoperation f:Sn→P∗(S) (or with Hinstead of Sif we wanted to make use of the distinction semihypergroup vs. hypergroup) defined by f(an 1)=[a1·. . . ·an | {z } n )≤={x∈S;a1·. . . ·an | {z } n ≤x}.(5) N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 152 The hypergroupoid would be an n-ary hypergroupoid and would be denoted in the former case by (S, ∗) and in the latter case by (S, f).∗ However, first of all we need to establish meaning of the very basic concepts used in (4) or (5). The result of the hyperoperation f(an 1) applied on elements a1, . . . , an,n > 2 is the upper end of a single element a1·. . . ·an | {z } n ∈S. (In further text we call such an element as generating the upper end.) Yet how does one obtain this single element? In other words, what is the arity of the single-valued operation ·? In a general case, ·may be a binary operation, an n−ary operation, or a j–ary operation for some special jsuch that 2 < j < n. In this paper we suppose that ·is a binary operation, i.e. that the product a1·. . . ·an | {z } n is an iterated binary operation. This is usually defined in such a way that for j≥1, n≥jwe denote by an ja sequence of elements ai,j≤i≤n and for the single-valued binary operation sfwe define two new operations sit l and sit rin the following way: sit l(an 1) = a1n= 1 sf(sit l(an−1 1), an)n > 1 and sit r(an 1) = a1n= 1 sf(an, sit r(an−1 1)) n > 1 Obviously, in a general case sit l(an 1)6=sit r(an 1). However, if the original binary operation sfis associative, then the two newly defined operations sit land sit r are equal and we may write sit instead. In the paper we will use the notation a1·. . . ·an | {z } n in the sense of sit(an 1). More precisely we should distinguish between sit l(an 1) and sit r(an 1) but this would be redundant because the construction we have been using and which we attempt to generalize, i.e. Lemma 3.1, assumes asociativity of the singlevalued operation. Remark 3.7. Notice that the decision on nature of a1·. . . ·an | {z } n has a number of implications. If contrary to our assumption one decides to consider this element as a result of an n–ary operation, then all theorems must be adjusted to work with n–ary quasi-ordered (semi)groups. These, however, must first be properly defined. Thus, from a certain point of view, our decision on the nature ∗Further on we will use the standard notation, i.e. define the n–ary hyperoperation using analogies of (5). Analogies of notation (4) will be used only at places where the explicit reference to the binary hyperoperation ∗makes the understanding more straightforward. N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 153 of a1·. . . ·an | {z } n is not only naturally following from the context but also easier and more convenient to work with. For details on iterated binary operations, cf. e.g. [18]. Remark 3.8. Just as we have considered the meaning of a1·. . . ·an |{z } n and discussed whether it is a result of an n–ary or an iterated binary single-valued operation ·, we may discuss the meaning of the symbol a1∗. . . ∗an | {z } n . Again, in a general case it could stand for both an n–ary or an iterated binary hyperoperation. Yet as has been suggested above, in the case of the hyperoperation we choose the n–ary option. 4 Associativity and commutativity First, discuss the issue of associativity and commutativity in n–ary hyperstructures defined by (5). Theorem 4.1. Let (S, ·,≤)be a quasi-ordered semigroup. n–ary hyperoperation f:Sn→P∗(S)defined by (5), i.e. as f(an 1)=[a1·. . . ·an | {z } n )≤={x∈S;a1·. . . ·an | {z } n ≤x}. is associative. Furthermore, it is commutative if the semigroup (S, ·)is commutative. 2 Proof. In order to prove associativity, we will modify the proof of [4], Lemma 1.6, p. 148, which shows that if we start with a partially ordered semigroup (S, ·) there holds a∗(b∗c) = (a∗b)∗c= [a·b·c)≤. First of all, suppose the following: x, y, ai∈S,i= 1, . . . , n + 1, x≤yand that (S, ·,≤) is a partially ordered semigroup. This implies that ai·x≤ai·y, x·ai≤y·aiand [ai·y)≤⊆[ai·x)≤, [y·ai)≤⊆[x·ai)≤for i= 1, . . . , n (and the same for any product of any number of elements of Sin position of ai– if we keep their order). Second, notice that obviously for all x∈Ssuch that an·an+1 ≤xthere is [a1·. . . ·an−1 | {z } n−1 ·x)≤⊆[a1·. . . ·an+1 | {z } n+1 )≤. This is easy to verify because the fact that y∈[a1·. . . ·an−1 | {z } n−1 ·x)≤is equivalent to the fact that a1·. . . ·an−1 | {z } n−1 · N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 154 x≤y. On the other hand, the fact that an·an+1 ≤xis equivalent to a1·. . . ·an+1 | {z } n+1 ≤a1·. . . ·an−1 | {z } n−1 ·x, which due to transitivity of the relation ≤means that a1·. . . ·an+1 | {z } n+1 ≤y, i.e. y∈[a1·. . . ·an+1 | {z } n+1 )≤. Naturally, it is not important whether we multiply by xfrom left or right, i.e. there is also [x·a3·. . . ·an+1 | {z } n−1 )≤⊆[a1·. . . ·an+1 | {z } n+1 )≤for all x∈Ssuch that a1·a2≤x. Then consider that the proof of Lemma 1.6 of [4] goes (using the above considerations for n= 2 and notation a, b, c instead of ai) as follows: a∗(b∗c) = [ x∈b∗c a∗x=[ x∈[b·c)≤ [a·x)≤= [a·b·c)≤∪[ x>b·c [a·x)≤= [a·b·c)≤ and similarly (a∗b)∗c=[ x∈[a·b)≤ [x·c)≤= [a·b·c)≤, which combined means that a∗(b∗c)=(a∗b)∗c=a∗b∗c. This can be denoted as f(a, f(b, c)) = f(f(a, b), c) or f(a1, f(a3 2)) = f(f(a2 1), a3) using the notation (5) for any triple of elements of S. Analogously we prove that f(a1, f(a4 2)) = f(f(a3 1), a4) = f(a4 1) for any quadruple of elements of Sas well as f(a1, f(a5 2)) = f(f(a4 1), a5) = f(a5 1) for any quintuple of elements of S. Thus for arity n= 3 we have that f(ai−1 1, f(ai+2 i), a5 i+3) = f(aj−1 1, f(ai+2 j), a5 j+3) for all i, j ∈ {1,2,3}, which means that associativity in 3–ary EL–hypergroupoids (S, f) is secured. Obviously, this consideration can be repeated for any higher arity n. Proving commutativity is rather simple: since the single-valued operation ·is commutative and as has been shown above also associative, then all permutations a1·. . . ·an | {z } n are equal. This means that all respective upper ends [a1·. . . ·an | {z } n )≤are equal because they are generated always by the same element. In other words, all permutations of the hyperoperation fare equal, i.e. the hyperoperation fis commutative. In [22] implications of the converse of Lemma 3.1 have been studied. The fact that commutativity of the binary hyperoperation implies commutativity of the single-valued operation is included already in Lemma 3.1. The same N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 155 fact on binary associativity was proved in [22] as Theorem 3.1. Notice that in both cases, the relation ≤must be partial ordering, i.e. not quasi-ordering only. This follows from the fact that the implication [a)≤= [b)≤⇒a=b(6) is valid only on condition of antisymmetry of the relation ≤, and the respective proofs make use of (6). For a counterexample of (6) used in the binary context of Lemma 3.1 cf. e.g. [22], Example 3.15. Let us now study the converse of Theorem 4.1. Theorem 4.2. Let (S, ·)be a non-trivial groupoid and ≤a partial ordering on Ssuch that for an arbitrary pair of elements (a, b)∈S2,a≤b, and for an arbitrary c∈Sthere holds c·a≤c·b,a·c≤b·c. Further define an n–ary hyperoperation f(also denoted by ∗) using notation (5) (or (4)). Then if the hyperoperation f(or ∗) is associative, then the single-valued operation ·is associative too. Furthermore, if the hyperoperation f(or ∗) is commutative, then the single-valued operation ·is commutative too. 2 Proof. The fact that the hyperoperation f(or ∗) is associative, means that all permutations f(ai−1 1, f(an+i−1 i), a2n−1 n+i) for an arbitrary i∈ {1,2, . . . , n}are equal, i.e. if an arbitrary element x∈Sbelongs to one of the permutations f(ai−1 1, f(an+i−1 i), a2n−1 n+i), it belongs to all other ones. Suppose an arbitrary x∈f(ai−1 1, f(an+i−1 i), a2n−1 n+i) for some i∈ {1,2, . . . , n}, e.g. for i= 1. This means that x∈f(f(an 1), a2n−1 n+1 ), i.e. using the ∗notation, x∈a1∗. . . ∗an | {z } n ∗an+1 ∗. . . ∗a2n−1 | {z } n−1 . This means that there exists an element x1∈a1∗. . . ∗an | {z } n such that x∈x1∗an+1 ∗. . . ∗a2n−1 | {z } n−1 . In other words, for these elements there holds that a1·. . . ·an | {z } n ≤x1and x1·an+1 ·. . . ·a2n−1 | {z } n−1 ≤x. Thanks to the properties assumed in the theorem this – when combined – means that (a1·. . . ·an | {z } n )·(an+1 ·. . . ·a2n−1 | {z } n−1 )≤x1·(an+1 ·. . . ·a2n−1 | {z } n−1 )≤x and thanks to assumed transitivity of the relation ≤we get that x∈[(a1·. . . ·an | {z } n )·(an+1 ·. . . ·a2n−1 | {z } n−1 ))≤.(7) N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 162 Theorem 5.16. Let (S, ·,≤)be a non-trivial quasi-ordered semigroup and (S, f)an n–ary EL–semihypergroup associated to it. If 0is the zero element of (S, f), then 0is the maximal element of (S, ≤). 2 Proof. From (14) in the definition of the zero element and from the definition of the hyperoperation fwe get that [x1·. . . ·xi−1 | {z } i−1 ·0·xi+1 ·. . . ·xn | {z } n−i )≤={0}(15) for every isuch that 1 ≤i≤nand for every (x1, . . . , xi−1, xi+1, . . . , xn)∈ Sn−1. Since the relation ≤is reflexive, there is x1·. . . ·xi−1 | {z } i−1 ·0·xi+1 ·. . . ·xn | {z } n−i ∈[x1·. . . ·xi−1 | {z } i−1 ·0·xi+1 ·. . . ·xn | {z } n−i )≤, which combined with (15) means that for a zero element 0 there must be x1·. . . ·xi−1 | {z } i−1 ·0·xi+1 ·. . . ·xn | {z } n−i = 0 for every isuch that 1 ≤i≤nand for every (x1, . . . , xi−1, xi+1, . . . , xn)∈Sn−1. Yet if this holds, (15) reduces to [0)≤={0}, which means that 0 is the maximal element of the relation ≤. Example 5.17. Since there are no maximal elements in (R,+,≤)there are no zero elements in (R, f)from Example 5.6. Example 5.18. If we want to describe zero elements in (P(S), f)from Example 5.7, we must concentrate on the only maximal element of (P(S),∪,⊆), i.e. on P(S)itself. We easily verify that it is a zero element of (P(S), f). Inverse elements in n–ary hyperstructures are studied e.g. in [2]. The property of having a unique inverse element required in [2] is taken over from the definition of canonical n–ary hypergroup included in [15]. Notice that canonical n–ary hypergroups are a special class of commutative n–ary hyperstructures (moreover, with the unique identity ehaving a certain further property), i.e. the definition of inverse elements included in [2], which has been taken over from [15], must be adjusted to a more general case. In the following text the notation perm{a1, . . . , an}stands for the set of all permutations of elements a1, . . . , an. Definition 5.19. Element x0of an n–ary hypergroup (H, f)is called an inverse element to x∈Hif there exists an identity e∈Hsuch that e∈f(perm{x, x0, e, . . . , e | {z } n−2 }) (16) N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 163 for every 1≤i≤n. Theorem 5.20. Let (H, f)be an n–ary EL–hypergroup associated to a quasiordered group (H, ·,≤). For an arbitrary x∈Hthere holds 1. if x0≤x−1, then x0is an inverse of xin (H, f), 2. if x0is an inverse of xin (H, f), then a≤x−1for all a∈perm{x0· e·. . . ·e | {z } 2(n−2) }, where x−1denotes the inverse of x∈Hin (H, ·)and eis some (unspecified) identity of (H, f). 2 Proof. Suppose that x∈H,x0∈Hare arbitrary and denote by the upper index −1the inverse in (H, ·). Finally, denote by uthe identity of (H, ·). Throughout the proof recall (5) on page 151 for the definition of the hyperoperation fusing the single-valued operation ·and the relation ≤. ad 1: If x0≤x−1, then also x0·x≤x−1·x=uand x·x0≤x·x−1=u. Moreover, we can multiply by the element uany number of times, or ”insert” it anywhere ”in between” xand x0or x0and xon the left side. Since according to Corollary 5.5 uis an identity of (H, f), we have that x0is an inverse of x. ad 2: Suppose that x0is an inverse of xin (H, f). This means that there exists an identity e∈Hsuch that (16) holds. This means that x·x0·e·. . . ·e | {z } arbitrary permutation of nelements ≤e When we multiply this by e·. . . ·e | {z } n−2 , we get x·x0·e·. . . ·e | {z } arbitrary permutation of x,x0and 2(n−2) instances of e ≤e·. . . ·e | {z } n−1 . However, from Theorem 5.2 and transitivity of the relation ≤we get that x·x0·e·. . . ·e | {z } arbitrary permutation of x,x0and 2(n−2) instances of e ≤u N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 164 which is equivalent to x0·e·. . . ·e | {z } arbitrary permutation of x0and 2(n−2) instances of e ≤x−1. It can be easily verified that commutativity / non-commutativity of the single-valued operation ·is not relevant in the last step. Remark 5.21. Notice that for arity n= 2 there is 2(n−2) = 0, i.e. Theorem 5.20 turns into an equivalence which enables us to describe the set of all inverses of an arbitrary x∈H(denoted as i(x)) in a far more elegant way by i(x)=(x−1]≤={x0∈G;x0≤x−1},(17) which has already been shown as [19], Theorem 3.9. Example 5.22. If we regard the hypergroup (R, f)from Example 5.6, we see that all a∈Rsuch that a≤ −xare inverses of an arbitrary real number x in (R, f). We also see that we might set e= 0 and Theorem 5.20 turns into equivalence. 6 A more complex example The hyperstructures (R, f) and (P(S), f) used to demonstrate the use of the above obtained results are quite simple and straightforward ones. Let us therefore conclude with a more complex example. Example 6.1. In paper [6] the authors deal with the relation of hyperstructures and homogeneous second order linear differential equations y00 +p(x)y0+q(x)y= 0,(18) such that p∈C+(I),q∈C(I), where Ck(I)denotes the commutative ring of all continuous real functions of one variable defined on an open interval I of reals with continuous derivatives up to order k≥0(instead of C0(I)the authors write only C(I)), and C+(I)denotes its subsemiring of all positive continuous functions. They denote the set of nonsingular ordinary differential equations (18) by A2, the pair of functions p, q by [p, q],D=d dxand the identity operator by Id. The notation L(p, q)is reserved for the differential operator L(p, q) = D2+p(x)D+q(x)Id, i.e. the notation L(p, q)(y) = 0 is the equation (18). The set LA2(I) = {L(p, q) : C2(I)→C(I); [p, q]∈C+(I)×C(I)}(19) N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 165 is the set of all such operators. Finally for an arbitrary r∈Rthe notation χr:I→Rstands for the constant function with value r. Proposition 1 of [6] states that if we define multiplication of operators by L(p1, q1)·L(p2, q2) = L(p1p2, p1q2+q1) (20) and if we define that L(p1, q1)≤L(p2, q2)if p1(x) = p2(x), q1(x)≤q2(x)for all x∈I, (21) then (LA2(I),·,≤)is a noncommutative partially ordered group with the unit element (identity) L(χ1, χ0). Using Lemma 3.1 and a further proof included in [6] we get that if we put L(p1, q1)∗L(p2, q2) = ={L(p, q)∈LA2(I); L(p1, q1)·L(p2, q2)≤L(p, q)}= (22) ={L(p1p2, q); q∈C(I), p1q2+q1≤q}, then (LA2(I),∗)is a (transposition) hypergroup ([6], Theorem 3).† Expand now the binary hyperoperation ∗defined in (22) for arity n= 3 and suppose the 3–ary hypergroupoid (LA2(I), f), where f(L(p1, q1), L(p2, q2), L(p3, q3)) = [L(p1, q1)·L(p2, q2)·L(p3, q3))≤,(23) for arbitrary operators, where ·is defined as (20) and ≤is defined as (21). According to Theorem 4.1 and Theorem 4.3, (LA2(I), f)is a noncommutative 3–ary hypergroup. According to Theorem 5.2, all operators L(p, q)such that p≡1,q(x)≤0for all x∈I, are identities of (LA2(I), f)and one can easily verify that also part 1 of the Theorem holds. In order to describe scalar identities of (LA2(I), f), Theorem 5.10 states that we have to examine operators L(a, b)such that for an arbitrary operator L(r, s)∈LA2(I)there simultaneously holds L(r, s) = L(a, b)·L(r, s)·L(a, b) L(r, s) = L(r, s)·L(a, b)·L(a, b) L(r, s) = L(a, b)·L(a, b)·L(r, s) If the operator L(a, b)does not have this property, then it is not a scalar identity. Yet since the result of the twice repeated multiplication in (23) is †Notice that if we do not restrict our considerations to positive continuous functions p and suppose that p(x)6= 0 for all x∈I, then we for sure know only that (LA2(I),∗) is a semihypergroup. However, it can be shown that even in this case it is a hypergroup. N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 166 L(p1, p2p3, p1p2q3+p1q2+q1), we have that the above conditions in fact mean that L(r, s) = L(a2r, arb +as +b) L(r, s) = L(a2r, rab +rb +s) L(r, s) = L(a2r, a2s+ab +b) which obviously holds for a≡1,b≡0only. Thus by Corollary 5.12 we get that there are no scalar identities in (LA2(I), f). Theorem 5.16 states that maximal elements of (LA2(I),≤)are the only potential zero elements of (LA2(I), f). However, no such elements exist in (LA2(I),≤), i.e. there are no zero elements in (LA2(I), f). As far as inverse elements of (LA2(I), f)are concerned, the operator L(1 p,−q p)is the single-valued inverse of L(p, q)∈LA2(I). Thus according to Theorem 5.20 all operators L(r, s)∈LA2(I), where r(x) = 1 p(x),s(x)≤ −q(x) p(x) for all x∈Iare inverses of an arbitrary operator L(p, q)in (LA2(I), f). 7 Conclusion This paper has contributed to the study of n–ary hyperstructures started only recently by [13, 17] and especially to the development of the theoretical background of hyperstructures constructed from quasi– or partially ordered semigroups, i.e. to one of classical areas in the hyperstructure theory. Some particular results obtained earlier in papers such as e.g. [19, 21, 22] can now be regarded as special cases of results obtained for n–ary hyperstructures in this paper. Thanks to this, some results included in e.g. [3, 4, 5, 14] may be studied or described more easily or from a different perspective. References [1] S. M. Anvariyeh, S. Momeni, n–ary hypergroups associated with n–ary relations, Bull. Korean Math. Soc. 50 (2013)(2), 507–524, http://dx.doi.org/10.4134/BKMS.2013.50.2.507. [2] R. Ameri, M. Norouzi, Prime and primary hyperideals in Krasner (m, n)–hyperrings, European J. Combin., 34 (2013), 379–390, http://dx.doi.org/10.1016/j.ejc.2012.08.002. [3] J. Chvalina, Commutative hypergroups in the sense of Marty and ordered sets, Gen. Alg. and Ordered Sets, Proc. Inter. Conf. Olomouc, (1994), 19– 30. N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 167 [4] J. Chvalina, Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups, Masaryk University, Brno, 1995 (in Czech). [5] J. Chvalina, ˇ S´arka Hoˇskov´a–Mayerov´a, A. D. Nezhad, General actions of hyperstructures and some applications, An. S¸t. Univ. Ovidius Constanta, 21(1) (2013), 59-82. [6] J. Chvalina, L. Chvalinov´a, Join spaces of linear ordinary differential operators of the second order, Folia FSN Universitatis Masarykianae Brunensis, Mathematica 13, CDDE – Proc. Colloquium on Differential and Difference Equations, Brno, (2002), 77–86. [7] I. Cristea, Several aspects on the hypergroups associated with n-ary relations, An. S¸t. Univ. Ovidius Constanta, 17(3) (2009), 99-110. [8] I. Cristea, S. Janˇci´c-Raˇsovi´c, Composition hyperrings, An. S¸t. Univ. Ovidius Constanta, 21(2) (2013), 81–94. [9] I. Cristea, M. S¸tef˘anescu, Hypergroups and n-ary relations, European J. Combin., 31(2010), 780–789, http://dx.doi.org/10.1016/j.ejc.2009.07.005. [10] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Dodrecht – Boston – London, 2003. [11] P. Corsini, Hyperstructures associated with ordered sets, Bul. of the Greek Math. Soc. 48 (2003), 7–18. [12] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, 1993. [13] B. Davvaz, T. Vougiouklis, n–ary hypergroups, Iran. J. Sci. Technol. Trans. A-Sci., 30(A2), 2006. [14] D. Heidari, B. Davvaz, On ordered hyperstructures, U.P.B. Sci. Bull. Series A, 73(2) (2011). [15] V. Leoreanu, Canonical n–ary hypergroups, Ital. J. Pure Appl. Math. 24 (2008). [16] V. Leoreanu-Fotea, P. Corsini, Isomorphisms of hypergroups and of n–hypergroups with applications, Soft. Comput. (2009), 13:985–994, http://dx.doi.org/10.1007/s00500-008-0341-9. [17] V. Leoreanu-Fotea, B. Davvaz, n–hypergroups and binary relations, European J. Combin., 29 (2008), 1207–1218, http://dx.doi.org/10.1016/j.ejc.2007.06.025. N–ARY HYPERSTRUCTURES CONSTRUCTED FROM BINARY QUASI–ORDERED SEMIGROUPS 168 [18] F. P. Miller, A. F. Vandsome, J. McBrewster, Iterated Binary Operation, Alphascript Publishing, 2010. [19] M. Nov´ak, Important elements of EL–hyperstructures, in: APLIMAT: 10th International Conference, STU in Bratislava, Bratislava, 2011, 151– 158. [20] M. Nov´ak, Potential of the ”Ends lemma” to create ring-like hyperstructures from quasi-ordered (semi)groups, South Bohemia Mathem. Letters 17(1) (2009), 39–50. [21] M. Nov´ak, Some basic properties of EL– hyperstructures, European J. Combin., 34 (2013) 446–459. http://dx.doi.org/10.1016/j.ejc.2012.09.005 [22] M. Nov´ak, The notion of subhyperstructure of ”Ends lemma”–based hyperstructures, Aplimat – J. of Applied Mathematics, 3(II) (2010), 237– 247. [23] I. G. Rosenberg, Hypergroups and join spaces determined by relations, Ital. J. Pure Appl. Math. 4 (1998), 93–101. Michal NOV´ AK, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick´a 8, 616 00 Brno, Czech Republic. Email: nov[email protected]