Classes of Universal Graphs Definable by Universal Graph Algebra Quasi-Identities

Universal graph algebras establish a connection between universal graphs (i.e. ni−ary relations) and universal algebras. A universal graph theoretic characterization of universal graph quasi-varieties, i.e. classes of universal graphs definable by universal graph quasi-identities, is given. The results give rise to a structure theorem of Birkhoff-type: A class of finite undirected universal graphs is a universal graph quasivarieties if and only if it is closed with respect to isomorphisms, induced subgraphs, finite disjoint unions and homogeneous subproducts. Some examples are also considered. Mathematics Subject Classification: 05C25, 08B15


Introduction
Graph algebras have been invented in [5] to obtain examples of nonfinitely based finite algebras.To recall this concept, let G = (V, E) be a (directed) graph with the vertex set V and the edge set E ⊆ V × V .Define the graph algebra A(G) corresponding to G to have the underlying set V ∪{∞}, where ∞ is a symbol outside V , and two basic operations, a nullary operation pointing to ∞ and binary one denoted by juxtaposition, given for u, v ∈ V ∪ {∞} by otherwise.
Universal graph algebras have been defined in [3] to obtain some basic properties of universal graph algebras.To recall this concept, let G = (V, E) be a multi graph with the vertex set V and the edge set E, an edge in E is an ordered pair of vertices (not necessarily distinct) of V and let τ = (n i ) i∈I , n i ≥ 2 be a sequence of natural number.A function f i : V n i → V is called an n i -ary operation defined on V , and is said to have arity n i .For convenient we label all edges of E to be distinct.Let E f i ⊆ V n i , where (v 1 , v 2 , ..., v n i ) ∈ E f i if and only if (v 1 , v 2 ), (v 2 , v 3 ), ..., (v n i −1 , v n i ) ∈ E and the labels of them are all different, further if e f i = (v 1 , v 2 , ..., v n i ) ∈ E f i and e f j = (u 1 , u 2 , ..., u n j ) ∈ E f j , then the intersection of the label set of (v 1 , v 2 ), (v 2 , v 3 ), ..., (v n i −1 , v n i ) and the label set of (u 1 , u 2 ), (u 2 , u 3 ), ..., (u n j −1 , u n j ) is an empty set.In this case G = (V, (E f i ) i∈I ) is called an universal graph of type τ = (n i ) i∈I , n i ≥ 2. Define the universal graph algebra A(G) of type τ = (0, (n i ) i∈I ), n i ≥ 2 corresponding to the universal graph G with the underlying set V ∪ {∞}, where ∞ is a symbol outside V , and basic operations, nullary operation pointing to ∞ and n i -ary operations f i , i ∈ I, given for elements of (V ∪ {∞}) n i , n i ≥ 2 by otherwise.
In [4] graph quasi-varieties had been investigated for finite undirected graphs in order to get graph theoretic results (structure theorems) from universal algebra via graph algebras.In the present paper these investigations are extended to finite undirected universal graphs.We ask for an universal graph theoretic characterization of universal graph quasi-varieties, i.e. the classes of universal graphs with can be defined by quasi-identities for their corresponding universal graph algebras.The answer is a theorem of Birkhoff-type, which uses graph theoretic closure operations.A class of finite undirected universal graphs is quasi-equational (i.e., an universal graph quasi-variety) if and only if it is closed with respect to isomorphisms, induced subgraphs, finite disjoint unions and homogeneous subproducts.

Preliminaries
We recall the following terminology and notations.Let G = (V, ( which is an initial point or a terminal point of an edge in E(G), E (G) be the set of all order pairs (u, v) ∈ V (G) × V (G) where u, v are initial and terminal points of an edge in Here, for a ∈ V (G), let a(j) denote the j-component: a = (a(j)) j∈I .Assume the sets V (G i ), i ∈ I to be pairwise disjoint (otherwise make the sets For class R of universal graphs, let IR, SR, P R, P f R, UR, U f R denote the classes of all isomorphic copies, induced subgraphs, direct products, finite direct products, disjoint unions, and finite disjoint unions of members of R, respectively.Let G d , G df and G uf be the class of directed universal graphs, the class of finite directed universal graphs and the class of finite undirected universal graphs, respectively.We omit the index f (= finite), if also infinite universal graphs are to be considered.For all universal graphs G under consideration, let G = (V, (E f i ) i∈I ) as defined in section 1 (introduction).The algebra Let W τ (X), τ = (0, (n i ) i∈I ), n i ≥ 2, be the set of all terms over the alphabet X = {x 1 , x 2 , x 3 , ...} defined inductively as follows: (i) every variable x i , i = 1, 2, 3, ..., and ∞ are terms, (ii) if t 1 , t 2 , ..., t n i are terms, then f i (t 1 , t 2 , ..., t n i ) is a term, where f i is an n i -ary operation.W τ (X) is the set of all terms which can be obtained from (i) and (ii) in finitely many steps.The leftmost variable of a term t is denoted by L(t) and the rightmost variable of a term t is denoted by R(t).The terms in which the symbol ∞ occurs are called a trivial terms.These terms evaluate to ∞ in every universal graph algebras.To every non-trivial term t of type τ = (0, (n i ) i∈I ), n i ≥ 2 we assign an universal graph G(t) = (V (t), (E f i (t)) i∈I ), where the vertex set V (t) is the set of all variables occurring in t and each E f i (t), i ∈ I is defined inductively by is a compound term and L(t 1 ), L(t 2 ), ..., L(t n i ) are the leftmost variables of t 1 , t 2 , ..., t n i respectively.Formally, we assign the empty graph φ, to every trivial term t.
For convenience we will introduce some more notations about universal graph in the following way: Let G = (V, (E f i ) i∈I ) be an universal graph.For any u, v ∈ V (G), we say that there exists a dipath from u to v in G if there exists a sequence of order pairs (a , then we say that G is a rooted universal graph with root u.We see that L(t) is a root of the universal graph G(t) and the pair (G(t), L(t)) is the rooted universal graph corresponding to t.
For a set Σ of equations and an assignment h : Let t ∈ W τ be a term.We write t(x i 1 , x i 2 , ..., x in ) if x i 1 , x i 2 , ..., x in are the variables which occur in t.Given a term t(x 0 , x 1 , ..., x n ), an universal graph G and a mapping h : , h(x n )) (the right hand side has to be computed in A(G)).Sometimes we call h an evaluation of the variables.

and only if the universal graph algebra A(G) has the following property: A mapping
, n i ≥ 2 be terms.Then the non-trivial equation s ≈ t is an identity in the class of all universal graph algebras iff either both terms s and t are trivial or none of them is trivial, A quasi-identity q is a finite set Σ = {s 1 ≈ t 1 , ..., s n ≈ t n } of equations together with an equation s ≈ t: we use the notation Σ → s ≈ t or A graph G satisfies the quasi-identities q, denoted by G |= q, if for every assignment h : V (q) → V (G) ∪ {∞} (V (q) denotes the set of variables occurring in q) the following implication holds:

This will be denoted by G |= h(Σ) → h(s) ≈ h(t) (or G |= h(q)
).Note that every equation s ≈ t can be considered as a quasi-identity because For a set R of universal graphs, let P h R (P hf R, resp.)denote the set of all homogeneous subproducts of families (finite families , resp.) of members of R.
If G is finite, this construction also gives a finite set I. Lemma 3.1.Let s and t be non-trivial terms and let B ⊆ i∈I G i be a homogeneous subproduct of (G i ) i∈I .Then for an assignment h :

Next, we suppose that h(s) = ∞. Then we have h(t) = ∞. Let h(s) = h(x) and h(t) = h(y). Hence h(x) = h(y), i.e. (h(x))(i) = (h(y))(i) for all
In the same manner, we can proof that if Lemma 3.2.Let s and t be non-trivial terms, G be a disjoint union of G 1 and G 2 , h : V (s)∪V (t) → V (G)∪{∞} be an assignment and let

Proof. First we show that h(s) = ∞, if h(V (s))
V (G 1 ) and h(V (s)) . Since G(s) is a connected universal graph.Hence, there is an edge e ∈ E(G(s)) which x and y occur.But since G 1 and G 2 are disjoint, there is no edge e ∈ h(V (s)) which h(x) and h(y) occur.Therefore h is not a homomorphism.We get h(s) = ∞.In the similar way we have

Lemma 3.3. For a class R of universal graphs we have IU
Proof. 1) Let B ⊆ i∈I G i be a homogeneous subproduct, G i ∈ R, i ∈ I and let q = (Σ → s ≈ t) ∈ QidR.We show that B |= q.Let h : V (q) → V (B) ∪ {∞} be some assignment such that B |= h(Σ).We have to show h(s) = h(t).Let h i be a composition of h and p i be the i-th projection.By Lemma 3.1, we obtain Again by Lemma 3.1, we get h(s) = h(t).Consequently,

an assignment and assume G |= h(Σ). We have to show G |= h(s) ≈ h(t). Define the assignments h
3) By 1) and 2), we have Now we are already to formulate the main theorem for quasi-varieties in G uf .Theorem 3.1.Let R be a non-empty subclass of G uf .Then Qvar G uf (R) = IU f P hf R.
Proof.Because of Lemma 3.3 and Proposition 3.1, it suffices to show Qvar G uf (R) ⊆ IU f P h R. Since every universal graph is the disjoint union of its connected components, it remains to show G ∈ IP h R for every given connected undirected universal graph G ∈ Qvar G uf (R).Let V (G) = {a 0 , ..., a n }.Consider the following set Σ of identities Then Σ is finite since G is finite.Obviously, under the canonical assignment g : x i → a i (i = 0, ..., n) we have G |= g(Σ).Thus for given a i , a j ∈ V (G), a i = a j , the quasi-identity q = Σ → x i ≈ x j does not hold in G. Since G ∈ Qvar(R), we have q / ∈ Qid(R), i.e.R q. Hence there is some R ∈ R with R q, i.e. there must be some assignment h : . By construction of Σ and connectedness of G, for every two variables x, y ∈ V (Σ) there is some sequence (z 1 , ..., x) ≈ z 1 , ..., (z m , ..., z m−1 ) ≈ z m , (y, ..., z m ) ≈ y of identities from Σ. Consequently, h Let us consider what happens if we want to treat also infinite universal graphs.Since every finitely generated subgraphs of a universal graph algebra is finite, we have Qvar(R) = Qvar(S f (R)) for a given class R of universal graphs, where S f R denotes the class of all finite induced subgraphs of members of R. Thus, for an arbitrary R ⊆ G u , we have  Many universal graph theoretic properties can be expressed as identities or quasi-identities.We mention here some examples which may be of universal graph theoretic interest, as well.
is the indicated class of undirected universal graphs.
Every universal graph variety is also a universal graph quasi-variety.The converse is not true, and we mention here some universal graph quasi-variety of undirected universal graphs without loops which are not universal graph varieties.

Conclusion and discussion
In this article, a universal graph theoretic characterization of universal graph quasi-varieties, i.e. the classes of universal graphs with can be defined by quasi-identities for their corresponding universal graph algebras, is given.We obtain a theorem of Birkhoff-type, which uses graph theoretic closure operations.A class of finite undirected universal graphs is quasi-equational (i.e., a universal graph quasi-variety) if and only if it is closed with respect to isomorphisms, induced subgraphs, finite disjoint unions and homogeneous subproducts.Finally, it is worth while to mention that the result can be a guideline for futher investigate in the directions of the equational logic for universal graph algebras, universal graph varieties, subvarieties, and subvarieties of universal graph varieties generated by universal graph algebras.

3Definition 3 . 1 .Remark 3 . 1 .Definition 3 . 2 .
Characterization of universal graph quasivarietiesFor a set Q of quasi-identities and a class R of universal graphs, letMod(Q) = {G ∈ G d | G |= Q} and Qid(R) = {q | q is aquasiidentity and R |= q}.We let Qvar(R) = Mod(Qid(R)) and for a given class G of universal graphs, Qvar G (R) = G ∩ Qvar(R).A class of this form is called a quasi-equational or a universal graph quasi-variety in G.In particular, Qvar G (R) is the universal graph quasi-variety generated by R in G.In general, universal graph quasi-varieties are not quasi-varieties in the usual universal-algebraic sense, e.g. the direct product of universal graph algebras is not a universal graph algebra.Clearly, Qvar G (R) consists of universal graphs from G whose universal graph algebras belong to the quasi-variety ISP P u R * (cf.[1] p. 219, Thm.2.25, R * = {A(G) | G ∈ R}).However, the most algebras in ISP P u R * are not universal graph algebras.Therefore it is reasonable to ask for an internal characterization of Qvar G (R) using operations on binary relations (= uiversal graphs) only to avoid ultraproducts and infinite universal graphs.A universal graph G is called a homogeneous subproduct of a family

Proposition 3 . 1 .
Let R be the class of universal graphs and G be a universal graph with |V (G)| ≥ 2. Then G ∈ IP h R if and only if for all a, b ∈ V (G) with a = b there exists an R ∈ R and a strong homomorphism φ

1 .
For a set Σ of identities and classes R and G of universal graphs letMod(Σ) = {G ∈ G d | G |= Σ}, Id(R) = {s ≈ t | R |= s ≈ t, s, t ∈ W τ (X)}, Var(R) = Mod(Id(R)) (the universal graph variety generated by R) V ar G (R) = G ∩ V ar(R) (the universal graph variety generated by R in G).
by Theorem 3.1.Moreover, we have G ∈ Qvar(R) if and only if S f ({G}) ⊆ Qvar G uf .This characterizes general universal graph quasi-varieties and our restriction to finite universal graphs was not essential.