On Definability of Universal Graphic Automata by Their Input Symbol Semigroups

Universal graphic automatonAtm(G,G) is the universally attracting object in the category of automata, for which the set of states is equipped with the structure of a graph G and the set of output symbols is equipped with the structure of a graph G preserved by transition and output functions of the automata. The input symbol semigroup of the automaton is S(G,G) = EndG×Hom(G,G). It can be considered as a derivative algebraic system of the mathematical objectAtm(G,G) which contains useful information about the initial automaton. It is common knowledge that properties of the semigroup are interconnected with properties of the algebraic structure of the automaton. Hence, we can study universal graphic automata by researching their input symbol semigroups. For these semigroups it is interesting to study the problem of definability of universal graphic automata by their input symbol semigroups— under which conditions are the input symbol semigroups of universal graphic automata isomorphic. This is the subject we investigate in the present paper. The main result of our study states that the input symbol semigroups of universal graphic automata over reflexive graphs determine the initial automata up to isomorphism and duality of graphs if the state graphs of the automata contain an edge that does not belong to any cycle.


INTRODUCTION
One of the main trends of the modern algebra is the generalized Galois theory, the basis of which was laid in the research of E. Galois and which is devoted to the study of mathematical objects by means of some derivative algebraic systems connected with the objects in a special way. Various algebraic systems, topological spaces, formal languages and many others were considered as initial mathematical objects, and automorphism groups and endomorphism semigroups of algebraic systems, homeomorphism groups and semigroups of continuous transformations of topological spaces, syntactic monoids of formal languages and many others were considered as derivative algebraic systems. Such studies for automorphism groups of algebraic systems, endomorphism semigroups of graphs, endomorphism rings of modules, and other derivative algebraic systems were very successfully carried out by B. I. Plotkin [1], A. G. Pinus [2,3], L. M. Gluskin [4,5], Yu. M. Vazhenin [6,7], A. V. Mikhalev [8] and many other algebraists. S. Ulam in his well-known book [9] has mentioned the problem of characterization of mathematical objects by their endomorphisms and automorphisms.
The generalized Galois theory also includes studies of structured automata, in which the systems of states and output symbols are objects of some category K (see, for example, review in [10]). In this case an automaton is considered as an initial mathematical object and the semigroup of input symbols is considered as a derivative algebraic system. For example, L. M. Gluskin, Yu. M. Vazhenin and other authors (see, review in [11]) investigated the endomorphism semigroups of graphs that can be considered as graphic semi-automata (i.e. automata without output symbols, for which the state systems are objects of the graph category Gr). In particular, Yu. M. Vazhenin in his studies [7] considered graphic semi-automata for graphs that contain an edge which does not belong to any cycle. The main result of the paper states that such graphic semi-automata are completely determined (up to isomorphism and duality of graphs) by their semigroups of input symbols.
In this paper we generalize this result for graphic automata, in which the systems of states and output symbols are objects of the graph category Gr. For any graphs G, G ′ , in the category of all graphic automata with the state graph G and the output symbol graph G ′ there exists the universally attracting object Atm(G, G ′ ) which is called a universal graphic automata over the graphs G, G ′ . The main result of the paper -Theorem 1 -states that for a reflexive graph G that contains an edge which does not belong to any cycle and a reflexive graph G ′ the automaton Atm(G, G ′ ) is completely determined (up to isomorphism and duality of graphs) by its input symbol semigroup.
The main result of this paper was announced in [12].

PRELIMINARIES
As a preliminary, we introduce some basic terminology from the theory of semigroups, the theory of automata and the theory of graphs that will be used and state the necessary notation. The reader interested in further details may consult, for instance, A. H. Clifford and G. B. Preston [13], B. I. Plotkin [10], A. M. Bogomolov and V. N. Salii [14], F. Harary [15].
Let X, Y be arbitrary sets and let ρ ⊂ X × Y be a binary relation. We put dom ρ = {x ∈ X : (∃y ∈ Y )(x, y) ∈ ρ}, ρ −1 = {(y, x) ∈ Y × X : (x, y) ∈ ρ}. By a mapping of a set X to a set Y, denoted ϕ : X → Y we mean a single-valued binary relation ϕ ⊂ X × Y such that dom ϕ = X. For an element x ∈ X, the image of x under ϕ is denoted by ϕ(x). For a subset A ⊂ X, let us denote ϕ(A) = {ϕ(x) | x ∈ A}.
A mapping ϕ : X → {x} denoted c x is called a constant mapping of a set X to an element x. A mapping of a set X to itself is called a transformation of X. The identity transformation of a set X is denoted by ∆ X . A one-to-one mapping ϕ : X → Y satisfying ϕ(X) = Y is called a bijection of X onto Y. In this case the inverse mapping For mappings ϕ : X → Y, ψ : Y → Z, the composition ϕψ : X → Z is defined by the formula (ϕψ)(x) = ψ(ϕ(x)) for x ∈ X. Denote the Cartesian product ϕ × ϕ = ϕ 2 , where ϕ 2 (x 1 , x 2 ) = (ϕ(x 1 ), (ϕ(x 2 )) for x 1 , x 2 ∈ X.
Following [14], by a graph we mean an algebraic structure G = (X, ρ), where X is a non-empty set and ρ is a binary relation ρ ⊂ X × X that is called an adjunct relation. The elements of X and ρ are called vertices and edges, respectively. The edge (x, x) ∈ ρ is called a loop. Vertices x, y ∈ X are called adjacent if (x, y) ∈ ρ or (y, x) ∈ ρ. If e = (x, y) is an edge then e is said to join the vertex x with the vertex y. Edges e 1 = (x 1 , y 1 ), e 2 = (x 2 , y 2 ) are called adjacent if y 1 = x 2 . A sequence of adjacent edges ❒àòåìàòèêà ✹✸ ➮çâ✳ Ñàðàò✳ óí✲òà✳ ❮îâ✳ ñåð✳ Ñåð✳ ❒àòåìàòèêà✳ ❒åõàíèêà✳ ➮íôîðìàòèêà✳ ✷✵✷✵✳ Ò✳ ✷✵✱ âûï✳ ✶ (x 0 , x 1 ), (x 1 , x 2 ), . . . , (x n−2 , x n−1 ), (x n−1 , x n ) is called a path from the vertex x 0 to the vertex x n . In case of equality x 0 = x n , the path is called a cycle. A graph G = (X, ρ) is called reflexive if its adjunct relation ρ is a reflexive binary relation, i.e. (x, x) ∈ ρ for any x ∈ X. Further, only reflexive graphs are considered.
For a graph G = (X, ρ) the graph G = (X, ρ −1 ) is called the dual graph of G.
We denote by Hom(G, G ′ ) the set of all homomorphisms of G to G ′ . Obviously, for ϕ is a bijection of X onto X ′ and the following condition holds: A homomorphism of a graph G = (X, ρ) to itself is called an endomorphism of G. We denote by End G the semigroup of all endomorphisms of G under the composition. Obviously, the identity transformation ∆ X of the vertex set X is an endomorphism of G. Moreover, for any x ∈ X, the constant mapping c x : X → {x} is an endomorphism of G.
For graphs G, G ′ , S(G, G ′ ) let us denote the semigroup End G×Hom(G, G ′ ) equipped with the following binary operation [10] (ϕ, ψ) · (ϕ 1 , ψ 1 ) = (ϕϕ 1 , ϕψ 1 ), We denote by Z(G, G ′ ) the set of all right zeros of the semigroup S(G, G ′ ) and by U (G, G ′ ) the set of all left identities of the semigroup S(G, G ′ ). These sets are respectively defined in the semigroup S(G, G ′ ) by the formulas of the semigroup theory Φ(x) = (∀y)(yx = x) and Ψ(x) = (∀y)(xy = y). By analogy with the lemmas 2.1-2.3 [16], it is easy to prove the following results.

Lemma 1. For any reflexive graphs
Then the formula of the semigroup theory defines the binary relation ε (G,G ′ ) on the semigroup S(G, G ′ ), such that the following statements hold: Following [10], by an automaton we mean a system A = (X, S, X ′ , δ, λ) consisting of a set of states X, a semigroup of input symbols S, a set of output symbols X ′ , a transition function δ : X × S → X and an output function λ : X × S → X ′ such that δ(x, st) = δ(δ(x, s), t) and λ(x, st) = λ(δ(x, s), t) for any x ∈ X and s, t ∈ S.
For every s ∈ S, we define the mappings δ s : X → X, λ s : X → X ′ by the formulas An automaton A = (X, S, X ′ , δ, λ) is said to be graphic if its sets X and X ′ are equipped with structures of graphs G = (X, ρ) and G ′ = (X ′ , ρ ′ ) such that for every s ∈ S the transition function δ s is an endomorphism of G and the output function λ s is a homomorphism of G to G ′ respectively. In this case we denote the automaton by A = (G, S, G ′ , δ, λ).

Proof. Obviously, 3) implies 1), 2). On the other hand, any isomorphisms
where (ϕ, ψ) ∈ S(G, G ′ ). Indeed, by the definition, the image f 2 (ϕ) = f −1 ϕf is an endomorphism of G 1 and the image (f × g)(ψ) = f −1 ψg is a homomorphism of G 1 to G ′ 1 . It is easy to verify that π is an isomorphism of the semigroup S(G, G ′ ) onto the semigroup S(G 1 , G ′ 1 ) such that the ordered triplet γ = (f, π, g) is an isomorphism of the automaton Atm(G, G ′ ) onto the automaton Atm(G 1 , G ′ 1 ). Similarly, any isomorphisms f, g of the graphs G, G ′ onto the dual graphs G 1 , G ′ 1 determine an isomorphism π : S(G, G ′ ) → S( G 1 , G ′ 1 ) by the formula π(ϕ, ψ) = (f 2 (ϕ), (f × g)(ψ)) (here (ϕ, ψ) ∈ S(G, G ′ )) such that the ordered triplet γ = (f, π, g) is an isomorphism of the automaton Atm(G, G ′ ) onto the automaton Atm( G 1 , G ′ 1 ). Hence, 1) implies 3). Let us now verify that 2) implies 1). Let π be an isomorphism of the semigroup S(G, G ′ ) onto the semigroup S(G 1 , G ′ 1 ). It is well-known that any isomorphism holds the true value of formulas of the semigroup theory. In particular, the isomorphism π preserves the true value of the formulas Φ(x), Ψ(x) and Σ(x, y). It follows that π maps the set Z(G, G ′ ) of right zeros of the semigroup S(G, G ′ ) onto the set Z(G 1 , G ′ 1 ) of right zeros of the semigroup S(G 1 , G ′ 1 ) and the set U (G, G ′ ) of left identities of the semigroup S(G, G ′ ) onto the set U (G 1 , G ′ 1 ) of left identities of the semigroup S(G 1 , G ′ 1 ). Moreover, by lemma 2 the Cartesian product π 2 maps the equivalence ε = ε (G,G ′ ) on the semigroup S(G, G ′ ) onto the equivalence ε 1 = ε (G 1 ,G ′ 1 ) on the semigroup S(G 1 , G ′ 1 ). In view of lemma 1, for any a ∈ X, a ′ ∈ X ′ , there exist a 1 ∈ X 1 , a ′ 1 ∈ X ′ 1 such that π(c a , c a ′ ) = (c a 1 , c a ′ 1 ). Hence, π maps the equivalence class ε(c a , c a ′ ) onto the equivalence class ε 1 (c a 1 , c a ′ 1 ). It follows that the formulas f (a) = a 1 and g a (a ′ ) = a ′ 1 determine the mappings f : X → X 1 and g a : X ′ → X ′ 1 (a ∈ X) such that the equality holds π(c a , c a ′ ) = (c f (a) , c ga(a ′ ) ).
It is easy to verify that f is a bijection of X onto X 1 and for every a ∈ X, g a is a bijection of X ′ onto X ′ 1 . Consider (ϕ, ψ) ∈ S(G, G ′ ) and for arbitrary a ∈ X we denote ϕ(a) = b, ψ(a) = d. ( By definition of S(G, G ′ ) for any y ∈ X ′ the equalities hold (c a , c y ) · (ϕ, ψ) = (c a ϕ, c a ψ) = (c ϕ(a) , c ψ(a) ).
It follows that the mapping π 1 = f 2 is a bijection of the set End G onto the set End G 1 . Moreover, it is obvious that f 2 preserves the composition. Hence, for any ϕ, ϕ 1 ∈ End G the equality π 1 (ϕϕ 1 ) = π 1 (ϕ)π 1 (ϕ 1 ) holds. Then the mapping π 1 is an isomorphism of the semigroup End G onto the semigroup End G 1 . By the condition of the theorem G is a reflexive graph with an edge (u 0 , v 0 ) ∈ ρ which does not belong to any cycle. It follows from [7] that the mapping f is an isomorphism of the graph G onto the graph G 1 or onto the dual graphs G 1 .
Suppose that f is an isomorphism of G onto G 1 . Hence (f (u 0 ), f (v 0 )) ∈ ρ 1 is an edge of the graph G 1 , which does not belong to any cycle. Next we show that, for any point a ∈ X the mapping g a is an isomorphism of the graph G ′ onto the graph G ′ 1 . It is easy to verify that for vertices x 0 , y 0 ∈ X ′ the condition (x 0 , y 0 ) ∈ ρ ′ is equivalent to the condition ψ 2 (u 0 , v 0 ) = (x 0 , y 0 ) for some homomorphism ψ ∈ Hom(G, G ′ ). Indeed, if ψ 2 (u 0 , v 0 ) = (x 0 , y 0 ) for some homomorphism ψ : G → G ′ then (x 0 , y 0 ) ∈ ρ ′ by the definition of graph homomorphism. On the other hand, let (x 0 , y 0 ) ∈ ρ ′ . Define a mapping ψ : X → X ′ by the following formula: We show that ψ is a homomorphism of the graph G to the graph G ′ . Let (s, t) ∈ ρ. If there exists a path from the vertex v 0 to the vertex s, then by the condition (s, t) ∈ ρ there exists a path from the vertex v 0 to the vertex t. Then by our definition of ψ the conditions ψ(s) = ψ(t) = y 0 hold. Since the graph G is reflexive, the conditions (ψ(s), ψ(t)) = (y 0 , y 0 ) ∈ ρ ′ hold. If there is no a path from the vertex v 0 to the vertex s, then by our definition of ψ the condition ψ(s) = x 0 holds. Since ψ(t) ∈ {x 0 , y 0 } and the graph G is reflexive, the condition (ψ(s), ψ(t)) ∈ ρ ′ holds. Thus, in any case the condition (s, t) ∈ ρ implies (ψ(s), ψ(t)) ∈ ρ ′ . Hence ψ ∈ Hom(G, G ′ ).

CONCLUSION
The results obtained shows that universal graphic automata over the graphs from a wide class of graphs are completely determined (up to isomorphism and the duality of ✹✽ ❮àó÷íûé îòäåë graphs) by their input symbol semigroups. Consequently, on the one hand, the characteristics of the input symbol semigroups of universal graphic automata must be defined by the characteristics of the automata, and on the other hand, the characteristics of these semigroups must characterize the automata to same extent. This approach allows us to study the properties of such automata by studying properties of their input symbols semigroups.