THE x-DIVISOR PSEUDOGRAPHS OF A COMMUTATIVE GROUPOID

The notion of a zero-divisor graph is considered for commutative groupoids with zero. Moufang groupoids and certain medial groupoids with zero are shown to have connected zero-divisor graphs of diameters at most four and three, respectively. As x ranges over the elements of a commutative groupoid A (not necessarily with zero), a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of A and vertices a and b are adjacent if and only if ab = x. These systems are completely characterized as being partitions of complete pseudographs Kn whose parts are indexed by the vertices of Kn. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it (categorically) isomorphic to the category of commutative groupoids, thereby combinatorializing the theory of commutative groupoids. Also, concepts of “congruence” and “direct product” that are compatible with those in the category of commutative groupoids are established for these systems of pseudographs. Mathematics Subject Classification (2010): 05C25, 18B99, 20N02


Introduction
There are several methods that exist for generating a groupoid (that is, a set endowed with a binary operation 1 ) from a graph.For example, cycle decompositions of complete graphs are used to produce groupoids in [17], and 1-factorizations of complete graphs are used to construct quasigroups (that is, cancellative groupoids A such that aA = Aa = A for every a ∈ A, i.e., for every a, b ∈ A there exist unique x, y ∈ A such that ax = ya = b) in [15] and [22].The zero-divisor graph of a commutative ring R was first defined in [5] to be the graph Γ(R) whose vertices are the elements of R such that distinct a and b are adjacent if and only if ab = 0.The 1 The present use of the term "groupoid" coincides with the work of B.A. Hausmann and O. Ore in [11] and seems to be standard among a majority of the related work in nonassociative algebra.
It should not be confused with the categorical convention in which a groupoid is defined to be a category such that every morphism is invertible.

THE x-DIVISOR PSEUDOGRAPHS OF A COMMUTATIVE GROUPOID
63 definition of Γ(R) was later revised in [2] to only include the nonzero zero-divisors of R, as the algebraic structure reflected by the adjacency relations involving 0 and nonzero-divisors are of little interest.The zero-divisor graph concept was extended to noncommutative rings in [21], commutative semigroups in [9], and partially ordered sets in [10].
In this paper, the notion of a zero-divisor graph is considered for commutative groupoids with 0. Furthermore, it is extended in a manner that permits any commutative groupoid to be viewed as a partition of a complete pseudograph, where a pseudograph is regarded as a graph without multiple edges that may (but is not required to) have some looped vertices.(Also, to avoid trivialities, the vertex-set V (G) of any pseudograph G will always be assumed nonempty.) A zero of a commutative groupoid A is an element 0 ∈ A such that a0 = 0 for every a ∈ A. Clearly a zero of A is unique.If A is a commutative groupoid with 0 then define the zero-divisor graph of A to be the graph Γ(A) whose vertices are the elements of {a ∈ A \ {0} | ab = 0 for some b ∈ A \ {0}} (i.e., the nonzero zero-divisors of A) and distinct vertices a, b ∈ A are adjacent if and only if ab = 0.
Diameter has been a topic of broad interest in the study of zero-divisor graphs (e.g., it has been investigated in zero-divisor graphs of various algebraic structures among [2], [3], [4], [9], [14], [18], [19], and [21], and other contributions have been made that are not reflected in the bibliography of this article).In particular, it was shown in [9] that the zero-divisor graph of a commutative semigroup is connected with diameter at most three.Therefore, the class of graphs that are realizable as zero-divisor graphs of commutative semigroups is relatively "small."In contrast with semigroups, Theorem 2.1 shows that every simple graph is realizable as the zero-divisor graph of a commutative groupoid with 0. In fact, for every simple graph G, the "freest" groupoid with 0 whose zero-divisor graph is G is given.
Nevertheless, it is natural to question the extent to which known results on diameter for zero-divisor graphs of semigroups can be generalized.An obvious approach is to examine more general notions of associativity.For example, a groupoid A is called a Moufang groupoid if (ab)(ca) = (a(bc))a for all a, b, c ∈ A. This condition has been considered extensively for quasigroups (e.g., a Moufang quasigroup was used in a construction of the notorious "monster group" in [7]), and has recently been studied in more general groupoids (see [20]).Theorem 2.2 addresses diameter for commutative Moufang groupoids with 0. Specifically, it is shown that zero-divisor graphs of commutative Moufang groupoids with 0 are connected with diameter at most four.
Another well known generalization of associativity is given by the "medial" property.A groupoid A is called medial (or entropic) if the equality (ab)(cd) = (ac)(bd) holds for every a, b, c, d ∈ A. Note that medial groupoids with identity are associative (let b be the identity).However, the weaker identity-type condition "a ∈ aA for every a ∈ A" does not guarantee associativity in a medial groupoid (e.g.let A = Q be the set of rational numbers under the operation ab := a + a + b + b, where "+" is the usual addition).Theorem 2.3 shows that if A is a commutative medial groupoid with 0 such that a ∈ aA for every a ∈ A then its zero-divisor graph is connected with diameter at most three.We reference [13] for other work on medial groupoids, where the stronger condition "A = aA for every a ∈ A" is considered.
Let x be an element of a commutative groupoid A. Attempting to extend the definition of a zero-divisor graph in order to arrive at a notion of an "x-divisor graph," one might choose the vertex-set to be {a ∈ A \ {x} | ab = x for some b ∈ A \ {x}}.For general x, however, algebraic relations involving x may not be clear, i.e., the solutions a ∈ A to the equation ax = x might be interesting.Thus, it seems constructive to allow every element of {a ∈ A | ab = x for some b ∈ A} to be a vertex.Actually, in the present article, we are interested in examining "x-divisor graphs" of a given groupoid A for several different elements x ∈ A simultaneously.
Comparisons of the relations between different graphs are more meaningful if the graphs have the same vertex-sets, so it will be beneficial to consider every element of A to be a vertex.Moreover, we will permit looped vertices, and define the x-divisor pseudograph of A to be the pseudograph Γ x (A) whose vertices are the elements of A such that (not necessarily distinct) a, b ∈ A are adjacent if and only if ab = x.
Let A be a commutative groupoid.For every a, b ∈ A, there exists a unique x ∈ A such that ab = x.Therefore, the complete pseudograph K | A| (i.e., the complete graph on |A| looped vertices) is given by the union as x ranges over A of the pseudographs Γ x (A).More generally, define a complete system of pseudographs (abbreviated by C.S.P.) to be a |I|-tuple ⟨G i ⟩ i∈I (where I is an indexing set) of pseudographs G i such that V (G i ) = I for every i ∈ I and, for all i, j ∈ I, there exists a unique k ∈ I such that i and j are adjacent in G k .Throughout, the C.S.P. ⟨Γ x (A)⟩ x∈ A given by a commutative groupoid A will be denoted by D(A), and will be referred to as the divisor pseudograph system of A.
Section 3 is devoted to developing a framework for an algebraic theory of C.S.P.s that is compatible with that of commutative groupoids by determining the "correct" definitions of a "C.S.P.-homomorphism" and its "image," a "quotient-C.S.P.," THE x-DIVISOR PSEUDOGRAPHS OF A COMMUTATIVE GROUPOID 65 and a "direct product" of C.S.P.s.The "homomorphisms" of C.S.P.s defined in Section 3 make the class of C.S.P.s into a category that is isomorphic to the category of commutative groupoids with usual groupoid-homomorphisms (Theorem 3.4(2)).
In particular, two commutative groupoids A and B are isomorphic if and only if the C.S.P.s D(A) and D(B) are "isomorphic" (Theorem 3.4(1)).A notion of "congruence" is given such that an equivalence relation ∼ on a commutative groupoid A is a congruence relation on A if and only if it induces a "congruence relation" on D(A), and in this case the C.S.P.s D(A/∼) and D(A)/∼ are "isomorphic" (in fact, they are equal; Theorem 3.7).Moreover, a "direct product" of C.S.P.s is defined such that, for a family {A j } j∈J of commutative groupoids, the C.S.P.s

D(
∏ j∈J A j ) and ∏ j∈J D(A j ) are "isomorphic" (in fact, they are equal), where ∏ j∈J A j is the usual direct product of groupoids (Theorem 3.9).Throughout, if groupoids A and B are isomorphic then we will write The pseudographs G and G ′ are isomorphic, written G ∼ = G ′ (there will be no risk of confusion with the notation for isomorphic groupoids), if there exists is also a homomorphism.A reference on graph theory can be found in [16], and [12] sufficiently addresses the relevant concepts from category theory.Surveys on zero-divisor graphs are given in [1] and [8], and a standard reference on groupoids is [6].

Zero-divisor graphs of commutative groupoids
While the more general x-divisor graph (an idea conceived from the notion of a zero-divisor graph) commands the primary focus of the present article, discussions on this subject seem premature in the absence of any literature on zero-divisor graphs of groupoids.The purpose of this section is to submit a brief introduction to these zero-divisor graphs by considering their diameters in contrast to those of the associative algebraic structures that have already been widely studied.We show that every simple graph is realizable as the zero-divisor graph of a commutative groupoid, and then close the section by presenting examples of results that show the "connected zero-divisor graph" phenomenon extends beyond associative algebra.
Given any nonempty set X, let F(X) be the free commutative groupoid on X, i.e., F(X) is the unique (up to isomorphism) commutative groupoid containing X satisfying the following universal property: If f : X → A is any function from X into a commutative groupoid A then there exists a unique groupoid-homomorphism ψ : F(X) → A that agrees with f on the set X. Note that F(X) can be regarded as the collection of all finite bracketed products of elements from X such that ab = xy for some a, b, x, y ∈ F(X) if and only if {a, b} = {x, y}.Let 0 be an element (not in F(X)) and define F 0 (X) = F(X) ∪ {0} to be the commutative groupoid with zero given by extending the operation on F(X) so that 0a = a0 = 0 for every a ∈ F 0 (X).
The following theorem shows that, for every simple graph G, there exists a "freest" commutative groupoid A with 0 whose zero-divisor graph is isomorphic to G.

Theorem 2.1. Let G be a simple graph. There exists a commutative groupoid
A with 0 such that Γ(A) = G, and A has the following universal property: Let is the identity homomorphism on A ′ (by setting B = A ′ and φ = ι A ′ and using the "unique groupoid-homomorphism" portion of the universal property applied to A ′ ) and Φ 2 Φ 1 is the identity homomorphism on A (by setting B = A and φ = ι A and using the "unique groupoid-homomorphism" portion of the universal property applied to A).
The next two theorems show that the well known fact that zero-divisor graphs of commutative semigroups are connected extends to commutative Moufang groupoids with 0, and commutative medial groupoids with 0 such that a ∈ aA.Throughout, the distance between two vertices a and b will be denoted by d(a, b), and if a 1 , . . ., a n ∈ A \ {0} (n is a positive integer) then the sequence (a 1 , . . ., a n ) will be called a pseudowalk if a i a i+1 = 0 for every i ∈ {1, . . ., n − 1}.Note that if Theorem 2.2.Let A be a commutative Moufang groupoid with 0. Then Γ(A) is connected with diameter at most four.
The following equalities are derived by using the Moufang property.
Relative to the previous result on Moufang groupoids, the case when A is medial such that a ∈ aA for every a ∈ A is proved more readily.The following example shows that the assumption "a ∈ aA for every a ∈ A" cannot be omitted in Theorem 2.3.

69
In particular, A is a commutative medial groupoid with zero element 0. However, Γ(A) is not connected (it is given by the complete graph on {2, 3, 4} together with the isolated vertex 1).
It is clear that the bound given in Theorem 2.3 is sharp (e.g., let A be the collection of all subsets of {1, 2, 3} under the operation ∩).However, the following question remains unanswered.
Question 2.5.Does there exist a commutative Moufang groupoid A with 0 such that Γ(A) has diameter four?

Complete systems of pseudographs
to G 2 is given by a function σ : ) is a homomorphism of pseudographs for every i ∈ I; in this case, the homomorphism of C.S.P.s will be denoted by σ : , where H j | σ denotes the subpseudograph of H j induced by σ(I).Let G 1 = ⟨G i ⟩ i∈I , G 2 , and G 3 = ⟨H j ⟩ j∈J be C.S.P.s, and suppose that σ : is a homomorphism of pseudographs for every i ∈ I. Hence, the composition If σ : I → J is injective, surjective, or bijective, then σ : G 1 → G 2 will be called injective, surjective, or bijective, respectively, and a bijective C.S.P.-homomorphism will be called an isomorphism.The complete systems G 1 and G 2 are isomorphic, Note that " ∼ =" is an equivalence relation on the collection of all C.S.P.s.Also, given C.S.P.s  would fail to be a homomorphism, where k ∈ I is the unique element such that σ −1 (x) ∈ N G k (σ −1 (y)).This yields the following observation.
We now begin to establish a dictionary between commutative groupoids and C.S.P.s.Proof.Note that Γ x (B)| σ = Γ x (σ(A)) for every x ∈ σ(A).Hence, the equalities is the divisor pseudograph system of the groupoid σ(A) (cf. Example 3.1).The "in particular" statement holds since D is surjective on objects and morphisms.
We now turn our attention to determining notions of a "quotient-C.S.P." and a "direct product" of C.S.P.s that are compatible with that of a quotient-groupoid and a direct product of groupoids.Incidentally, as we shall see, such concepts turn out to be closely related to the usual notions of a quotient-pseudograph and a direct product of pseudographs (as defined in [16]).We begin with the concept and i ∈ X (i.e., the notation "H X " is taking the place of "Γ X " in the definition of a quotient-system).Since D(I /∼) is the divisor pseudograph system of the groupoid I /∼, it is sufficient to prove the second statement of the theorem.

Φ
then there exists a unique groupoid-homomorphism Φ : A → B that agrees with φ on V (G), i.e., the following diagram is commutative.Moreover, A is unique (up to isomorphism).Proof.Let I ⊆ F 0 (V (G)) be the set consisting of 0, all elements that have a bracketed factor of the form ab for some a, b ∈ V (G) with a ∈ N G (b), and all elements that have a bracketed factor of the form cc for some isolated vertex c ∈V (G) (i.e., N G (c) = ∅).Define an equivalence relation ∼ on F 0 (V (G)) by x ∼ y if and only if either x = y or x, y ∈ I. Set A = {a | a ∈ F 0 (V (G))},where a denotes the equivalence class of a.It is straightforward to check that ∼ is a congruence relation on F 0 (V (G)), and so the operation ab = ab makes A into a commutative groupoid with zero element 0 = I.By identifying an element a ∈ V (G) with a in A, it is clear that Γ(A) = G.Let B be a commutative groupoid with 0 and φ : V (G) → V (Γ(B)) ⊆ B a graphhomomorphism.Set Φ(0) = 0, and for every finite bracketed product x of elements THE x-DIVISOR PSEUDOGRAPHS OF A COMMUTATIVE GROUPOID 67 from V (G) let Φ(x) be the finite bracketed product in B obtained from x by replacing every factor a ∈ V (G) of x with φ(a).The resulting function Φ : A → B is a well defined (since φ is a graph-homomorphism) groupoid-homomorphism such that Φι A = φ, and the verification of the remaining assertions in the theorem are routine: Φ is unique because A is generated by (the image of) V (G), and then the uniqueness of A follows since, for any other A ′ with Γ(A ′ ) = G that satisfies the universal property, there exist homomorphisms Φ 1 : A → A ′ (by setting B = A ′ and φ = ι A ′ and applying the universal property to A) and Φ 2 : A ′ → A (by setting B = A and φ = ι A and applying the universal property to A ′ ) such that Φ 1 Φ 2 ab = 0 then d(a, b) ≤ 1.Also, if ay = 0 then d(a, b) ≤ 2, and if xy = 0 then d(a, b) ≤ 3. Therefore, assume that ab, ay, and xy are nonzero.Suppose that a(xy) = 0.If b(xy) = 0 then d(a, b) ≤ 2, so assume b(xy) ̸ = 0.If (ay)b = 0 then d(a, b) ≤ 3 since (a, xy, ay, b) is a pseudowalk by (1), so assume (ay)b ̸ = 0.If (xy) ( (ay)b ) = 0 then (a, xy, (ay)b, b) is a pseudowalk by (2), and if (xy) ( (ay)b ) ̸ = 0 then (a, xy, (xy) ( (ay)b ) , b(xy), b) is a pseudowalk by (3), (4), and (5).Thus, d(a, b) ≤ 4. Henceforth, assume that a(xy) ̸ = 0. Suppose that (ay)b = 0.If (ay) ( a(xy) ) = 0 then (a, a(xy), ay, b) is a pseudowalk by (6), so d(a, b) ≤ 3. On the other hand, if (ay) ( a(xy) ) ̸ = 0 then (a, a(xy), (ay) ( a(xy) ) , ay, b) is a pseudowalk by (6), (7), and (8).Hence, d(a, b) ≤ 4. Therefore, assume that (ay)b ̸ = 0. Suppose that a(ay) = 0. Then (a, ay, ab, (ay)b, b) is a pseudowalk by (9), (10), and (2), and hence d(a, b) ≤ 4. Thus, assume that a(ay) ̸ = 0.The cases when either a(xy) = 0, (ay)b = 0, or a(ay) = 0 have been exhausted.But if the elements a(xy), (ay)b, and a(ay) are nonzero then (a, a(xy), a(ay), (ay)b, b)

Remark 3. 3 . ( 1 )Proposition 3 . 5 .
Let A be a commutative groupoid.It has already been observed that D(A) = ⟨Γ x (A)⟩ x∈ A is a C.S.P. Conversely, let G = ⟨G i ⟩ i∈I be a C.S.P.Note that setting ij = k if and only if i, j, k ∈ I with i ∈ N G k (j) makes I into a well defined (by the uniqueness of k) commutative groupoid JOHN D. LAGRANGE σ : A → B, it is clear that D is faithful.But it is observed in Remark 3.3(1) that D is bijective on objects, and it follows that D is an isomorphism.The next result supports the given definition of the "image" of a C.S.P.-homomorphism.If σ : A → B is a homomorphism of commutative groupoids then D(σ(A)) = D(σ)(D(A)).In particular, any homomorphic image of a C.S.P. is a C.S.P.

Remark 3 . 6 .Figure 2 .Theorem 3 . 7 .
Figure 2. If ∼ is the equivalence relation on {a, b, c} whose equivalence classes are given by a = {a} and b = c = {b, c} then G /∼ is not a C.S.P.