On path hypercompositions in graphs and automata

The paths in graphs define hypercompositions in the set of their vertices and therefore it is feasible to associate hypercompositional structures to each graph. Similarly, the strings of letters from their alphabet, define hypercompositions in the automata, which in turn define the associated hypergroups to the automata. The study of the associated hypercompositional structures gives results in both, graphs and automata theory.


Introduction
An operation or composition in a non-void set H is a function from HuH to H while a hyperoperation or hypercomposition is a function from HuH to the power set P (H) of H.An algebraic structure that satisfies the axioms: i. a(bc) = (ab)c for every a,b,c H (associativity) ii.aH = Ha = H for every a H (reproductivity) is called group if « » is a composition, and hypergroup [6] if « » is a hypercomposition [13].A set H endowed with a hypercomposition "" is called hypergroupoid if xyz for all x, y in H, otherwise it is called partial hypergroupoid.If A and B are non-empty subsets of H, then AB signifies the union  for all a,b,c H (mixed associativity) [7] A transposition hypergroupoid is a hypergroupoid which satisfies the axiom [5]: b\a c/d z implies ad bc z A commutative transposition hypergroup is called join hypergroup or join space [5,8].
In general graph is a set of points called vertices connected by lines called edges.A path in a graph is a sequence of no repeated vertices v 1 , v 2 , …, v n , such that , are edges in the graph.The length of a path is the number of edges that it uses.A graph is said to be connected if every pair of its vertices is connected by a path.A directed graph (or digraph) is a graph, where the edges have a direction associated with them.A degenerate edge of a graph which joins a vertex to itself, also called a self-loop or loop.Multiple edges are two or more edges that connect the same two vertices.The term multigraph refers to a graph which has multiple edges between nodes.A directed graph (or digraph) is a graph, where the edges have a direction associated with them.A simple graph (or strict graph), is an unweighted, undirected graph containing no graph loops or multiple edges.A tree T is a simple, connected graph with no cycles.A spanning tree of a connected graph is a tree whose vertex set is the same as the vertex set of the graph, and whose edge set is a subset of the edge set of the graph.
An automaton A is a collection of five objects (Ȉ, S, į, s o , F) where Ȉ is the alphabet of input letters (a finite nonempty set of symbols), S is a finite nonvoid set of states, s o is the start (or initial) state, an element of S, F is the set of the final (or accepting) states, a (possibly empty) subset of S and į is the state transition function with domain SuȈ and range S, in the case of a deterministic automaton (DFA), or P (S), in the case of a nondeterministic automaton (NDFA).Ȉ * denotes the set of words (or strings) formed by the letters of Ȉ -closure of Ȉ-and ȜȈ * signifies the empty word.Ȉ * under the concatenation of words is a monoid, with neutral element Ȝ, since Ȝx=xȜ=x for all x in Ȉ * .Moreover Ȉ * becomes a hyperingoid under the b-hyperoperation: x+y={x, y} for all x, y in Ȉ * [20].Given a DFA A, the extended state transition function for A, denoted į * , is a function with domain SuȈ * and range S defined recursively as follows: i. į * (s,a) = į(s,a) for all s in S and a in Ȉ ii.į * (s,Ȝ) = s for all s in S iii.į * (s,ax) = į * (į(s,a),x) for all s in S, x in Ȉ * and a in Ȉ. P. Corsini [2], M. Gionfriddo [4], Nieminen [21,22], I. Rosenberg [23], M. De Salvo et al. [24] and others studied hypergroups associated with graphs.G. G. Massouros [14][15][16][17][18] and after him J. Chvalina [1] studied hypergroups associated with automata.Moreover, in [15] G. G. Massouros introduced the path hypercomposition in graphs and subsequently Ch.G. Massouros and G. G. Massouros introduced in [9] another type of path hypercomposition in graphs and some relevant hypercompositions in automata.

The path hypercompositions in Graphs
In the set V of the vertices of a tree, a hypercompostion "" has been introduced in [9] as follows: for each two vertices x, y in V, xx=x and xy is the set of all vertices which belong to the path that connects vertex x with vertex y.Since tree is an undirected graph, this hypercomposition is commutative.Furthermore, this hypercomposition is a closed hypercomposition.Therefore: Proposition 4. If V is the set of the vertices of a tree T, then V = x/x, for each x in V.
The set <x,y> = x/y xy y/x, where xzy are two vertices of T , is called the line of T which is defined by In [9] it is proved that the lines of T are convex sets.Moreover the following important theorem it is proved in [9]: Theorem 1.If V is the set of the vertices of a tree T, then (V, ) is a join space.
It is known that any connected graph has at least one spanning tree and that there exist algorithms which find such trees.Hence any graph can be endowed with the join space structure through its spanning trees.
Theorem 2. Let G be a connected graph and T a spanning tree of G.The set of the vertices of the graph becomes a join space if for all vertices x, y of G , the hypercomposition xx T y is the set of all vertices which belong to the path that connects vertex x with vertex y in T .
Since a graph may have more than one spanning trees, more than one join spaces can be associated to a graph.
Next, define in the set V of the vertices of a tree T a hypercomposition "´ VXFK WKDW for each two vertices x, y in V, xy consists of all the internal vertices which belong to the path that connects vertex x with vertex y, that is, if , , are edges in a path connecting the vertices x and y, then

<
. This hyper- composition is an open hypercomposition.It is obvious that , V < is a partial hypergoupoid, since the result of the hypercomposition of two successive vertices is void.If the above hypercomposition is introduced in a simple connected graph, then it is possible to exist more than one paths connecting two vertices x, y of the graph.Hence if , , are edges in a path which connects the vertices x and y, then ^, ,..., v n v v x y , V < will be a hypergroupoid if and only if for any two vertices x and y of V there exists a path from x to y of length greater or equal to 2.
The Boolean domain B = {0, 1} becomes a semiring under the addition 0 + 1 = 1 + 0 = 1 + 1 = 1, 0 + 0 = 0 and the multiplication 0 0 = 0 1 = 1 0 = 0, 1 1 = 1.This semiring is called a binary Boolean semiring.A Boolean matrix is a matrix with entries from the binary Boolean semiring.A square Boolean matrix is called total if all its entries are equal to 1 [10].The adjacency matrix of a graph on n vertices is an n x n Boolean matrix A = (a i,j ) in which the entry a i,j equals to 1, if there is an edge from vertex i to vertex j and equals to 0 if there is no edge from vertex i to vertex j.Through the adjacency matrix a binary relation ȡ can be defined in in the set V of the vertices as follows: In [11] the following theorem has been proved:

The path hypercompositions in Automata
In [14][15][16][17][18][19] it has been shown by G. Massouros, that the set of the states of an automaton, equipped with different hypercompositions, can be endowed with the structure of the hypergroup.The hypergroups that have derived in this way were named attached hypergroups to the automaton.Up to this point several kinds of attached hypergroups have introduced in order to describe the structure and the operation of the automata with the use of tools from the Hypercompositional Algebra.Among them there are: i. the attached hypergroups of the order, and ii. the attached hypergroups of the grade.These two kinds of hypergroups have also been used for the minimization of the automata.
Moreover, in [15] another hypergroup, which derived through a different consideration of the hypercomposition, has been attached to the set of the states of an automaton.Due to its definition this hypergroup was named by G. Massouros attached hypergroup of the paths and it has led to a new proof of Kleene' s theorem.Furthermore, in [16], the attached hypergroup of the operation has been attached to the automaton.Apart from the other results, this hypergroup can indicated all the states in which an automaton can be found after the tclock pulse.
Hereafter two hypercompositions will be presented which are defined through the strings of letters from the alphabet of the automaton.Let A be the automaton (Ȉ, S, į, s o , F).If x be a word in Ȉ * , then: Prefix(x)={yȈ * ° yz=x for some zȈ * } and Suffix(x)={zȈ * ° yz=x for some yȈ * } Let s be an element of S. Then I s = {xȈ * ° į * (s o ,x)=s} and P s = {s i S° s i =į * (s o ,y), yPrefix(x), xI s } Considering the automaton as a directed graph, P s is the set of the states which appear in all possible paths connecting the start state s o with the state s.Since ȜȈ * the states s o and s are in P s .In the set of the states of A we introduce the hypercomposition s + q = P s P q for all s, q S (1) This hypercomposition is commutative, thus the two induced hypercompositions coincide and so we have: ^q s q S, s P s/ q = q\ s = r S| P P s P

°® °r if if
It is proved that [9]: Proposition 5.The set S endowed with the hypercomposition ( 1) is a join hypergroup.
The other hypercomposition is defined as follows: s + q = P s P q for all s, q S (2) Since s o P r for all r S, the results of hypercomposition (2) are always non void sets.Moreover this hypercomposition is commutative, thus the two induced hypercompositions coincide and so we have: q q S, s P s/ q = q\ s = , s P °® °if if It has been proved that [9]: Proposition 6.The set S endowed with the hypercomposition ( 2) is a join semihypergroup.

Proposition 3 .
If the hypercomposition in a hypergroupH is right or left open, then all its elements are idempotent .Two induced hypercompositions (the left and the right division) derive from the hypercomposition of the hypergroup[6], i.e. a/b = {x H ~ a xb} and b\a = {y H ~ a by} When "" is commutative, a/b = b\a.Consequences of the axioms (i) and (ii) are: i. ab z , for all a,b H. ii.a/b z and a\b z , for all a,b H. iii. the nonempty result of the induced hypercompositions is equivalent to the reproductive axiom.iv.(a/b)/c = a/(cb), c\(b\a) = (bc)\a, (b\a)/c = b\(a/c),

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201 Let V be the set of the vertices of a graph G.