Abstract When we assume that the input-set of an automaton without output is a semihypergroup instead of a monoid, we talk about quasi-multiautomata. Even though cartesian composition of quasi-automata is a commonly used concept, the cartesian composition of quasi-multiautomata has not been successfully constructed yet. In our paper we show that the straightforward transfer of the deffinition into the multivariate context fails. We suggest two possible solutions of this problem.

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Cartesian composition and the problem of generalizing the MAC condition to quasi-multiautomata

A functorial passage from the category of automata without outputs and their homomorphisms into the category of preorder hypergroups and strong homomorphisms based on the concept of inertial relation extended into apreording of a statě set is ušed for the describing of some basic properties of automata. Further, the relational hypergroup product is treated in connection with products of automata. 1991 Mathematics Subject Classification: 20N20, 68Q70 In the páper [16] there are described constructions of some hyperstructures on sets of words formed from the given input alphabets and on the statě sets of corresponding automata. Using the hyperoperation x -f y = {x,y} on the set A* of words over a given alphabet A and the binary operation of concatenation, the authors of [16] háve introduced a notion of a hyper-ringoid, and the equivalence on A* determined by the equality of lengths of words has served them for a construction of the so called strengthen join hypergroup. Further in [16] there are defined various hyperoperations on the statě set of a deterministic and non-deterministic acceptor. One of them is defined in such a way that the result of the hyperproduct applied on a pair of stat.es is the union of blocks of the corresponding states within the grade equi valence. Here, by a grade of a statě s is called the set grád s of all the words transfering the statě 5 into the set of hnal states and two states s\, $2 are said to be grade equivalent if grád S\ = grád $2Such a defined hyperoperation on the statě set of an automaton creates a join hypergroup. Relationships between properties of automata and their corresponding hyperstructures are not treated in this páper [16]. Other current papers devoted to the mentioned topič in some other directions are [15], [17]. In this contribution we use a similar idea as it is described above (originally ušed in [16] independent ly within [7]) and to any automaton without output (in the sense e.g. [1], [2], [8], [9], [19]) we assign a commutative hypergroup (a quasiordering hypergroup — [4], [6], determined by the Warner quasi-ordering — [23]) in such a way that we get a functorial passage from the category of automata with the samé input alphabet into the category of commutative hypergroups and their strong homomorphisms. Some basic properties of automata will be described in terms of hypergroups. Finally a relational product of quasi-order hypergroups corresponding to the heterogenous product of automata will be introduced. 106 J. Chvalina, L. Chvalinová Recall first, that a hypergroup (in the sense of Marty) i.e. a pair (II, o), where H 7-: 0 and o : H x H —> exp' H (= exp II \ {0}) is an associative hyperoperation ((a o b) o c = a o (b o c), where A o I? = U a ° ^ f° r adnonempty subsets A, (a^eAxB B of H) satisfying the reproduction axiom ao I7 = II = IIoafor any a G II, is said to be a quasi-ordering hypergroup if for any a, 6 E II we have a G a = a , a o b = a U b . If moreover a = b implies a = b for any pair a, b £ H then (H, o) is called an ordering hypergroup ([6]). It is evident that a (quasi-)ordering hypergroup is commutative and extensive (which means {a, b} C a o b for any pair a, b G II). Let (H, o) be a hypergroup, I\ be a nonempty subset of II which is multiplicatively closed, i. e. K o K C K. If (K, o) satisfies the reproduction axiom then it is called a subhypergroup of the hypergroup (H, o) cf. [4], Definition 7, p. 8. It is to be noted that a quasi-ordering hypergroup can be defined in this spacesaving form: A quasi-ordering hypergroup is a hypergroupoid (I1,o) (i.e. H / 0, o : II x II -» exp H) such that a G a C a, a o b = a U b for any a,b £ H. It is easy to see that (H, o) satisfies the -reproduction axiom. Indeed, a o H C H for any a G II and for any x G H we have x£aoxCaoH, thus H C a o H for every a G 1I. Hence (H, o) is a commutative quasi-hypergroup. We show that the hyperoperation o is also associative. Define a binary relation r C H x H by x r y if and only if y G x, i. e. # = r(#) for any x £ H. Since the hypergroupoid (H, o) is extensive, the relation r is reflexive and for xry, yrz we have y £ x, z £ y which implies z £ x = x thus xrz, consequently r = r. Now, suppose a,b,c £ H are arbitrary elements. Then a o (b o c) = a o (r(b) U r(c)) = a o r ({b, c}) = Uar€r({6,c» a ° x = = («) U U«€r({6,c}) r M = r(a) U r ({b, c}) = r ({a, b, c}) . Similarly (a o b) o c = c o (b o a) = r ({a, b, c}), hence a o (b o c) = (a o b) o c and we have (H, o) is an extensive commutative hypergroup. As a motivating example for our consideration take the Peano algebra (N, a) of all positive integers with the successor function a : N —r N and define a hyperoperation o : N x N —•> exp' N in this way: For m.n G N denote a = min{m, n} and m o n = {x £ N, 3 k £ N or k = 0 : cr (a) = x} = {x £ N; a < x} . Then (N, o) is an ordering hypergroup. On the other hand, if we define a binary relation r C N x N by the rule (m^n) £ r iff n o m o n = n, then r is an ordering on the set N identical with the usual one <. The above mentioned simple relationship between quasi-ordering hypergroups and quasi-ordered sets can be also described as there follows: State hypergroups of automata 107 For any quasi-ordering hypergroup (H, o), the pair (H, r ) , where arb iff a o b o a = a , is a quasi-ordered set and if for any quasi-ordered set (X, r) we pu t x o y = r (#) U r(y) ( = {t; x* 7*/} U {.s; yr s}) then (X, o) is a quasi-ordering hypergroup. Now, let A* he the free monoid of words over an arbi t rary (nonempty) a lphabet A] suppose e s tands for the empty word. In accordance with [2], [3], [8], [9] and other pub l ications, by an au toma ton we mean a tr iad A = (S\ A, 8), where S\ A are arbi t rary sets (A ^ 0), which are called — m the given order — a set of s ta tes (or a s ta te set) , a set of input symbo ls (or an input a lphabet) and 8 : S x A* —> S is a mapping called a t ransi t ion function', which satisfies these two condit ions: 8(s, e) = .s for any s ta te s G S, ( the identity axiom), 8(s,ab) = 8(8(s,a),b) for any s ta te .s G .5 and any pair of words a,b G A* (the homomorphism axiom). It is to be noted tha t the transi t ion function 8 : S x A* —> S is a usual extension of the next-s ta te function 8A : S x A —» S. An a u t o m a t o n A] = (S\,A,8\) is said to be a subau tomaton of an au toma ton A = (S\A,8) if S\ C S, 8(s,a) G S\ for any s ta te .s G S\ and any word a G A* and further 8] = 8 t ,S' x A*, i. e. o\ is a restriction of the t ransi t ion function 8 on Si x A*. For l / T C 5 we denote £ (T ,A*) = {<*(*, a ) ;* G Tya G A*} . (In [1], [2] this operator is denoted only by 8, i .e . 8(T) s tands for the above set in the mentioned papers) . If T is a singleton, e .g . T = {t}, we write S(t, A*) instead of S({t}, A*). Let A = (S, A, 8) be an au toma ton . We define a binary hyperoperat ion on the s ta te set S of A by sot = 6(s,A*)US{tiA*) for any pair of s ta tes .s, t G S. Since A* contains the empty word e, we have ,s, t G -s o /, thus s o t / 0 and commuta t iv i ty of this hyperoperat ion is also evident. Further, .s = 8(s, A*) thus s o l = .s U l and .s = . s o . s 2 = s o o > , A * ) = U t ^ ^ = .s U <J(<J(s, A*), A*) = .s U {S(8(s, a),b);a£A*,beA*} = = sU{8(s,ab);abeA*A* = A*} = s U {S(s, c); c £ A*} = .s U £(,s, A*) = = .s U .s = .s , i .e . .s G -s = .s for any s ta te s G S. Consequently, the hyperoperat ion o is associative and we get t ha t (S, o) is a quasi-ordering hypergroup. The same also follows from the fact, t ha t .sol = {u G S\ (s, u) G r or (t, u) G r } = r( .s)Ur(/) , where r is the t ransi t ive cover of the inertial relation v on the s ta te set of A((s,t) G v if J(.s, a) = t for some input symbol a £ A) which has been used and studied by M . W . W a r n e r in [23]. The jus t defined quasi-ordering hypergroup (s,o) will be called the s ta te hypergroup of the au toma ton A = (S,A,8) and denoted also M(A). 108 J.Chvalina, L. Chvalinová Recall that a nonempty subautomaton B = (T, A,S) (automata with empty state sets will not be considered) of an automaton A = (5, A,S) is said to be separated if S(S \ T, A*) fl T = 0. An automaton is called connected if it does not possess any separated proper subautomaton. Definition. We say that a commutative hypergroup (H, *) is an inner disjoint product of its subhypergroups (Hi, *), (H2, *), if H = Hi * H2, Hi fl H2 = 0. Definition. A commutative hypergroup (H, *) is said to be inner irreducible if for any pair Hi, H2 of its subhypergroups such that Hi*H2 = H we have HiflH2 70, i. e. if (H, *) is not any inner disjoint product of some pair of its subhypergroups. T h e o r e m 1. An automaton A = (S, A,S) is connected if and only of its state hypergroup (,$', o) is inner irreducible. P R O O F : Suppose the automaton A is connected and (Hi ,o), (H2,o) are subhypergroups of the state hypergroup (S, o) such that Hi o H2 = S. For any state s G Hi and any word a G A* we have S(s,a) eS(s,A*) = s o s G Hi, thus (Hi, A, Sx) with Si = S f Hi x A* is a subautomaton of the connected automaton A. Then S(S \ Hi, A*) O Hi 70 hence there exists a pair of states (t, s) £ Hi x (S \ Hi) and a word a £ A* such that t = S(s,a) CS(s,A*) = sos. From the equality Hx oH2 = S there follows the existence of a pair (u, v) G Hi x H2 such that s G u o v = £(u, A*) U S(v, A*) . Since u G Hi, <$(tx, A*) = ti o tx € # 1 , s € .$' \ Hi and <J(v, A*) = v o v C H2, we have ^ G s o 5 C [JX£V0V y£vov xoy=(vov)o(vov) = v ov = = v ov = v = v £ H2, thus t G Hi OH2, hence Hi DH2 70 which means that the state hypergroup (5, o) of the automaton A is inner irreducible. Now suppose (S, o) is inner irreducible and simultaneously the automaton A is disconnected. Then s(s\Si,A*)nSi = 0 for some subautomaton (Si, A,Si) of the automaton A. Denoting Hi = S\, H2 = S \ S\ then from the equality Si(Si,A*) = S\ there follows (Hi,o) is a State hypergroups of automata 109 subhypergroup of the hypergroup (S, o). Let s,t G H2. Since S(H2,A*) C\ H\ = 0,

State hypergroups of automata

This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and the hypergroup. Next, it presents the several types of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides the ones it was initially equipped with, along with their fundamental properties. Furthermore, it analyzes and studies the various subhypergroups that can be defined in hypergroups in combination with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract group theory, and demonstrates the different techniques and special tools that must be developed in order to achieve results on hypercompositional algebra.

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An Overview of the Foundations of the Hypergroup Theory

The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the paths of the theory of Formal Languages, Automata and Geometry. This paper presents the course of development from the hypergroup, as it was initially defined in 1934 by F. Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleen’s Theorem, determine the order and the grade of the states of an automaton, minimize it and describe its operation. The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry, like Helly’s Theorem, Kakutani’s Lemma, Stone’s Theorem, Radon’s Theorem, Caratheodory’s Theorem and Steinitz’s Theorem. This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries, like projective geometries and spherical geometries.

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Hypercompositional Algebra, Computer Science and Geometry

The hypergroup (as defined by F. Marty), being a very general algebraic structure, was subsequently quickly enriched with additional axioms. One of these is the transposition axiom, the utilization of which led to the creation of join spaces (join hypergroups) and of transposition hypergroups. These hypergroups have numerous applications in geometry, formal languages, the theory of automata and graph theory. This paper deals with transposition hypergroups. It also introduces the transposition axiom to weaker structures, which result from the hypergroup by the removal of certain axioms, thus defining the transposition hypergroupoid, the transposition semi-hypergroup and the transposition quasi-hypergroup. Finally, it presents hypercompositional structures with internal or external compositions and hypercompositions, in which the transposition axiom is valid. Such structures emerged during the study of formal languages and the theory of automata through the use of hypercompositional algebra.

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G. G. Massouros

Papers

Hypercompositional Structures in the Theory of the Languages and Automata.