Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups

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Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups John D. LaGrange Division of Natural Science and Mathematics, Lindsey Wilson College, Columbia, KY, USA

a r t i c l e

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Article history: Received 13 September 2013 Received in revised form 12 January 2015 Available online xxxx Communicated by A. Knutson MSC: 05C25; 06A12; 20M14

a b s t r a c t Let V (G) be the set of vertices of a simple connected graph G. The set L1 (G) consisting of ∅, V (G), and all neighborhoods N (v) of vertices v ∈ V (G) is a subposet of the complete lattice L(G) (under inclusion) of all intersections of elements in L1 (G). In this paper, it is shown that L1 (G) is a join-semilattice and L(G) is a Boolean algebra if and only if G is realizable as the zero-divisor graph of a meetsemilattice with 0. Also, if L1 (G) is a meet-semilattice and L(G) is a Boolean algebra, then G is realizable as the zero-divisor graph of a join-semilattice with 0. As a corollary, graphs that are realizable as zero-divisor graphs of commutative semigroups with 0 that do not have any nonzero nilpotent elements are classified. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Given a (multiplicative) commutative semigroup S with 0, the zero-divisor graph Γ(S) of S is the (undirected) graph whose vertices are the nonzero zero-divisors of S such that two distinct vertices x and y are adjacent (via a single edge) if and only if xy = 0. The concept of a zero-divisor graph was introduced by I. Beck in [6]. In Beck’s zero-divisor graph, the vertices were the elements of a commutative ring, and his study was focused on colorings. Of course, the zero-divisor relations involving 0 or any nonzero-divisor are fully understood. Therefore, when zero-divisor graphs are studied in order to illuminate algebraic structure, it is customary to restrict the set of vertices to only include the nonzero zero-divisors of S. This approach was first taken by D.F. Anderson and P.S. Livingston in [5] while studying commutative rings, and was extended to commutative semigroups with 0 by F.R. DeMeyer, T. McKenzie, and K. Schneider in [11]. Surveys on zero-divisor graphs are provided in [2] and [8]. Recently, the zero-divisor graph concept has been extended to posets (that is, to partially ordered sets; see [12,14,17,19]). Let P be a poset with a least element 0. Given a subset ∅ = A ⊆ P, define A∧ = {x ∈ P | x ≤ a for every a ∈ A}. The zero-divisor graph of P, denoted by Γ(P), is the (undirected) graph whose vertices   are the nonzero elements of Z(P) = x ∈ P | {x, y}∧ = {0} for some 0 = y ∈ P such that two vertices E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2016.01.012 0022-4049/© 2016 Elsevier B.V. All rights reserved.

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x and y are adjacent (via a single edge) if and only if {x, y}∧ = {0} (equivalently, the infimum x ∧ y of {x, y} exists in P and is 0). Although zero-divisor graphs of certain posets have been considered earlier (e.g., zero-divisor graphs of Boolean algebras were considered in [4]), zero-divisor graphs of general posets were first studied by R. Halaš and M. Jukl in [14]. As with Beck’s paper, they allowed every element of P to be a vertex, and they were mainly interested in colorings. The present definition was used by D. Lu and T. Wu in [19]. Zero-divisor graphs of posets were extended to quasiordered sets (i.e., sets endowed with a relation that is reflexive and transitive) by R. Halaš and H. Länger in [15], and the case when P is a lattice was later studied by E. Estaji and K. Khashyarmanesh in [12]. The present paper focuses on semilattices with a least element 0. Sections 2 and 3 extend the Introduction by providing brief expositions on the essential topics that will be further developed in this article. Sections 4, 5, and 6 continue the investigations from [17] and [19]. By imposing additional conditions to a characterization of zero-divisor graphs of posets given in [19], the zero-divisor graphs of meet-semilattices are completely determined, and sufficient conditions are provided to determine whether a graph is realizable as the zero-divisor graph of a join-semilattice (Theorems 4.4 and 5.1). As a corollary, zero-divisor graphs of commutative semigroups without nonzero nilpotent elements are classified (Corollary 1.2). Along the way, a graph-theoretic analogue of the usual ring-theoretic annihilator is developed, and is used to organize graph-theoretic criteria in an algebraic form (e.g., see Section 3). The set of vertices of a graph G will be denoted by V (G). Two simple graphs G1 and G2 are called isomorphic, written G1 ∼ = G2 , if there exists a bijection ϕ : V (G1 ) → V (G2 ) such that two vertices x and y are adjacent in G1 if and only if ϕ(x) and ϕ(y) are adjacent in G2 . Let G be a simple graph. If v ∈ V (G), then let cG (v) = {w ∈ V (G) | w is adjacent to v}. Given any A ⊆ V (G), define cG (A) = V (G) if A = ∅, and otherwise let cG (A) = ∩{cG (v) | v ∈ A}. When there is no risk of confusion, the set cG (A) will be denoted by c(A). (The choice in notation follows from the fact that “c” is a complementation on the lattice L(G) = {c(A) | A ⊆ V (G)} under inclusion; see Section 3.) The neighborhood in G of a set ∅ = A ⊆ V (G) is the set NG (A) = ∪{cG (v) | v ∈ A}. When A = {a1 , . . . , an } is a finite set, then we will write NG (A) = NG (a1 , . . . , an ). Of course, NG (a) = cG (a) for any vertex a of G. For consistency, we will always write cG (a) instead of NG (a). Also, when there is no risk of confusion, the set NG (A) will be denoted by N (A). Let L1 (G) = {c(a) | a ∈ V (G)} ∪ {∅, V (G)} ⊆ L(G). The following corollary is proved in Section 5. Corollary 1.1. Let G be a simple connected graph with 2 ≤ |V (G)| < ∞. Then the following statements are equivalent. (1) (2) (3) (4) (5) (6)

L1 (G) is a lattice, and L(G) is a Boolean algebra. L1 (G) is a meet-semilattice, and L(G) is a Boolean algebra. L1 (G) is a join-semilattice, and L(G) is a Boolean algebra. G∼ = Γ(S) for some lattice S. G∼ = Γ(S) for some meet-semilattice S. G∼ = Γ(S) for some join-semilattice S.

The next corollary summarizes several of the main results of this paper, and is proved in Section 6. Corollary 1.2. Let G be a simple connected graph with |V (G)| ≥ 2. Then the following statements are equivalent. ∼ Γ(S) for some reduced commutative semigroup S. (1) G = (2) G ∼ = Γ(S) for some commutative Boolean semigroup S.

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(3) G ∼ = Γ(S) for some meet-semilattice S. (4) For every a, b ∈ V (G), either a ∈ cG (b), or there exists an x ∈ V (G) that satisfies conditions (1) and (2) of Theorem 4.4. (5) L1 (G) is a join-semilattice, and L(G) is a Boolean algebra. (6) L1 (G) is a join-semilattice, and G ∼ = Γ(P) for some poset P. / c(b). (7) L1 (G) is a join-semilattice, and c(a) ∨ c(b) = V (G) for every a, b ∈ V (G) with a ∈ Throughout this paper, all posets and semigroups are assumed (usually implicitly) to have a 0, and zero-divisor graphs are implicitly assumed to have nonempty vertex-sets. By definition, all zero-divisor graphs are simple graphs (i.e., no loops or multiple edges). Also, it is well known that zero-divisor graphs of semigroups are connected [9, Theorem 1(2)], and that zero-divisor graphs of posets are connected with at least two vertices [19, Proposition 2.1 and Theorem 3.1]. Therefore, while the characterizations of zero-divisor graphs in this paper are usually stated under the assumption that graphs are simple and connected (e.g., see Corollary 4.6), note that generality is not lost. For references on lattices and semigroups, see [13] and [18]. For a reference on graph theory, see [7]. 2. Algebraic annihilators Recall that a semigroup (respectively, ring) S is a Boolean semigroup (respectively, Boolean ring) if x2 = x for every x ∈ S. Note that the infimum operation ∧ makes any meet-semilattice into a commutative Boolean semigroup. Conversely, for any commutative Boolean semigroup S, the relation that is given by defining x ≤ y in S if and only if xy = x makes S into a meet-semilattice with x ∧ y = xy for every x, y ∈ S [18, Proposition 1.1.1]. In this case, if S has 0, then xy = 0 if and only if x ∧ y = 0. In particular, the zero-divisor graph of a commutative Boolean semigroup coincides with the zero-divisor graph of its corresponding meet-semilattice. To see how this relationship can be extended to more general semigroups, let S be a commutative semigroup with 0. Given any a ∈ S, let annS (a) = {x ∈ S | xa = 0}. In general, if ∅ = A ⊆ S, then the set annS (A) = ∩{annS (a) | a ∈ A} is called the annihilator of A in S. For every a ∈ S, define [a] = {y ∈ S | annS (y) = annS (a)}. Note that the relation ∼ defined by a ∼ y if and only if [a] = [y] is an equivalence relation on S. Also, the operation [a][b] = [ab] is well-defined, and makes the set SE = {[a] | a ∈ S} into a commutative semigroup with a zero given by [0]. The zero-divisor graph of SE was considered in [20] by S.B. Mulay in the context of commutative rings. It was later studied in [3,8,21]. A semigroup with 0 (respectively, ring) S is called a reduced semigroup (respectively, reduced ring) if S does not contain any nonzero nilpotent elements (equivalently, a2 = 0 for every 0 = a ∈ S). For example, every Boolean semigroup is reduced. If S is a reduced commutative semigroup, then it is straightforward to check that SE is a commutative Boolean semigroup. In particular, SE has a zero-divisor graph that coincides with that of its corresponding meet-semilattice. To exploit the relationship between the zero-divisor graph of S and that of the meet-semilattice corresponding to SE , we introduce some graph-theoretic concepts. Let G be a simple graph. An equivalence relation is defined on V (G) by declaring two vertices v and w to be equivalent if and only if cG (v) = cG (w). The reduction G of G is then defined as the graph whose vertices are the equivalence classes in V (G) such that two vertices v and w are adjacent in G if and only if v ∈ cG (w). For a reduced commutative semigroup S, Theorem 6.1 proves Γ(SE ) ∼ = Γ(S). This theorem will serve as the bridge between results on semilattices and results on semigroups (e.g., see the proof of Corollary 1.2). 3. Graph-theoretic annihilators For nonzero elements a and b of a reduced commutative semigroup S, the containment a ∈ annS (b) is equivalent to a ∈ cΓ(S) (b). Therefore, given a simple graph G, the set cG (A) (where A ⊆ V (G)) will

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be called the (graph-theoretic) annihilator of A in G. As we shall witness throughout this investigation, it turns out that this notion of “annihilator” generalizes some of the familiar properties of the usual (algebraic) annihilator annS (A). Let R be a reduced commutative ring with 1 = 0. A Boolean algebra is a complemented distributive lattice with at least two elements. Recall that the set A(R) = {annR (A) | ∅ = A ⊆ R} of annihilator ideals in R forms a complete Boolean algebra under inclusion [18, Proposition 2.4.2]. To examine connections between the algebraic and graph-theoretic annihilators, let G be a simple graph. Define L(G) = {c(A) | A ⊆ V (G)}. Then L(G) is a complete lattice under inclusion with least element c(V (G)) = ∅ and greatest element c(∅) = V (G). In particular, given any set {c(Ai )}i∈I ⊆ L(G), it is straightforward to check that   ∧{c(Ai ) | i ∈ I} = ∩{c(Ai ) | i ∈ I} = c ∪ {Ai | i ∈ I} and ∨{c(Ai ) | i ∈ I} = ∩{c(A) | A ⊆ V (G) and ∪i∈I c(Ai ) ⊆ c(A)}   = c ∪ {A | A ⊆ V (G) and ∪i∈I c(Ai ) ⊆ c(A)} . If a, b ∈ V (G), then it follows that c(a) ∧ c(b) = c({a, b}) and c(a) ∨ c(b) = c({x ∈ V (G) | N (a, b) ⊆ c(x)}). Moreover, L(G) is complemented. In fact, [17, Proposition 3.1] shows that c(X) is a complement of X for every X ∈ L(G). Let R be a reduced commutative ring with 1 = 0 that contains nonzero zero-divisors. It is straightforward to check that |V (Γ(R))| ≥ 2, and the mapping from A(R) to L(Γ(R)) defined by  annR (A) →

∅, if A contains a nonzero-divisor in R c(A \ {0}), otherwise

is an isomorphism of posets. Motivated by the fact that A(R) (and hence L(Γ(R))) is a Boolean algebra, simple graphs G such that L(G) is a Boolean algebra are classified in [17, Theorem 3.6]. For example, given a simple connected graph G with |V (G)| ≥ 2, it is proved that L(G) is a Boolean algebra if and only if there exists a poset P such that G ∼ = Γ(P) [17, Theorem 3.3]. In the present investigation, the two notions of “annihilator” are further connected. Given a reduced commutative ring with 1 = 0, recall that A1 (R) = {annR (r) | r ∈ R} is a join-semilattice (under inclusion) with annR (1) = {0}, annR (0) = R, and annR (r) ∨ annR (s) = annR (rs) for every r, s ∈ R (indeed, A1 (R) is isomorphic to the dual of the meet-semilattice RE ; cf. [3, Lemma 3.5]). In fact, it follows from [3, Lemma 3.5] that the correspondence given by [r] → annR (r) is an order-reversing isomorphism between posets RE and A1 (R). Also, it is straightforward to check that the mapping from A1 (R) to A1 (RE ) = {annRE ([r]) | [r] ∈ RE } defined by annR (r) → annRE ([r]) is an isomorphism of posets, and hence the mapping given by [r] → annRE ([r]) is an order-reversing isomorphism between the posets RE and A1 (RE ). Let G be a simple graph. A graph-theoretic analogue of the set A1 (R) is given by defining L1 (G) = {c(a) | a ∈ V (G)} ∪{∅, V (G)} ⊆ L(G). Note that L1 (G) is a subposet of the lattice L(G) under inclusion. In contrast to A1 (R), however, L1 (G) need not be a semilattice. For example, consider the graph G constructed by assigning a pendant vertex to each of two distinct vertices of a complete graph on four vertices (e.g., see Fig. 3(b)). Then L1 (G) is not a semilattice (e.g., the annihilators of the two pendant vertices have no supremum, and the annihilators of the two non-pendant vertices which are not adjacent to either of the pendant vertices have no infimum). Moreover, if L1 (G) is a lattice, then it need not be a sublattice of L(G) (e.g., if G is the complete graph on three vertices, say V (G) = {a, b, c}, then c(a) ∧ c(b) = ∅ in L1 (G) but c(a) ∧ c(b) = c({a, b}) = {c} in L(G)).

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Fig. 1. The poset P is not a semilattice, but L1 (Γ(P)) is a lattice.

Analogous to the order-reversing isomorphism between RE and A1 (RE ) presented above, it is easy to check that, if P is a poset with greatest element 1, then ψ : P → L1 (Γ(P)) defined by ψ(0) = V (Γ(P)), ψ(a) = ∅ for every a ∈ P \ Z(P), and ψ(a) = c(a) for every a ∈ Z(P) \ {0} is surjective and satisfies ψ(b) ⊆ ψ(a) whenever a ≤ b. However, in contrast to the order-reversing isomorphism between RE and A1 (RE ), it can happen that a and b are incomparable for some a, b ∈ P with ψ(b) ⊆ ψ(a). In particular, P need not be a meet-semilattice (respectively, join-semilattice) whenever L1 (Γ(P)) is a join-semilattice (respectively, meet-semilattice). For example, if P is the poset whose Hasse diagram is given in Fig. 1(a), then P is not a semilattice (e.g., x∨y and a∧b do not exist in P), but it is easy to check that L1 (Γ(P)) is a lattice. Conversely, Example 5.3 provides a join-semilattice (in fact, a lattice) S such that L1 (Γ(S)) is not a meet-semilattice. Let R be a reduced commutative ring with 1 = 0 that contains nonzero zero-divisors. It is straightforward to check that the isomorphism between A(R) and L(Γ(R)) defined above restricts to an isomorphism between A1 (R) and L1 (Γ(R)) given by

annR (a) →

⎧ ⎪ ⎨

∅, if a ∈ R \ Z(R) . V (Γ(R)), if a = 0 ⎪ ⎩ c(a), otherwise

The fact that A1 (R) is a join-semilattice, and A(R) is a Boolean algebra (and hence L1 (Γ(R)) is a joinsemilattice, and L(Γ(R)) is a Boolean algebra) motivates one to examine when L1 (G) is a join-semilattice, and L(G) is a Boolean algebra for a simple connected graph G. It turns out that, for a simple connected graph G with |V (G)| ≥ 2, the poset L1 (G) is a join-semilattice, and the lattice L(G) is a Boolean algebra if and only if G ∼ = Γ(S) for some meet-semilattice S (Corollary 4.6). Furthermore, Theorem 5.1 shows that, if the poset L1 (G) is a meet-semilattice, and the lattice L(G) is a Boolean algebra, then G ∼ = Γ(S) for some join-semilattice S. However, in contrast to Corollary 4.6, the converse of Theorem 5.1 fails by Example 5.3. Moreover, Example 4.7 provides a graph G such that L1 (G) is a join-semilattice (in fact, a lattice), but G  Γ(P) for every poset P (i.e., L1 (G) is a join-semilattice, but L(G) is not a Boolean algebra). General implications of L1 (G) being a semilattice (in particular, implications which are independent of L(G)) remain open, and are not addressed in the present investigation. 4. Zero-divisor graphs of meet-semilattices There are several papers that have contributed answers regarding which graphs are realizable as zerodivisor graphs (e.g., [4,9,10,16,19] contain such results, and there have been many other contributions which

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Fig. 2. The graph G in (a) is isomorphic to the zero-divisor graphs of both posets P1 and P2 , whose Hasse diagrams are given in (b) and (c), respectively. Given the graph G, the construction defined in the proof of [19, Theorem 3.1] yields the poset P1 , and the construction defining P(G) yields the poset P2 . Observe that P2 is a lattice, while P1 is not a semilattice since x ∨ y and x ∧ y do not exist in P1 .

are not reflected in the bibliography of the present article). In [16, Theorem 3.1], a characterization (independent of any algebraic information) is given for graphs G such that G ∼ = Γ(R) for some Boolean ring R with 1 = 0. In [19, Theorem 4.3], graphs having a finite clique number that are realizable as zero-divisor graphs of meet-semilattices are characterized. These results are extended below in Theorem 4.4, where zero-divisor graphs of commutative Boolean semigroups are classified (in the context of meet-semilattices). The following lemmas address conditions on the neighborhood of a set of two vertices. Lemma 4.1 considers the case when the two vertices are adjacent, and it is valid in every simple graph. Lemma 4.2 is well known, and it describes neighborhoods of nonadjacent vertices in zero-divisor graphs of posets (in fact, [19, Theorem 3.1] shows that it characterizes zero-divisor graphs of posets; see the discussion that follows Lemma 4.2). Lemma 4.1. Let G be a simple graph. If a, b ∈ V (G) such that a ∈ c(b), then N (a, b)  c(x) for every x ∈ V (G). Proof. If N (a, b) ⊆ c(x) for some x ∈ V (G), then a ∈ c(b) ⊆ N (a, b) ⊆ c(x). Thus, x ∈ c(a) ⊆ N (a, b) ⊆ c(x), which contradicts that G is a simple graph. Therefore, N (a, b)  c(x) for every x ∈ V (G). 2 Lemma 4.2. Let G be a simple graph such that G ∼ / c(b), = Γ(P) for some poset P. If a, b ∈ V (G) such that a ∈ then there exists an x ∈ V (G) such that N (a, b) ⊆ c(x). / c(b), Proof. Without loss of generality, suppose that G = Γ(P) for some poset P. Let a, b ∈ V (G). If a ∈ then there exists a 0 = x ∈ {a, b}∧ . Thus, x ∈ V (G), and it is clear that N (a, b) ⊆ c(x). 2 In [19, Theorem 3.1], it is shown that a simple connected graph G with at least two vertices is the zero-divisor graph of a poset if and only if, for any nonadjacent vertices a and b of G, there exists a vertex x of G such that N (a, b) ⊆ c(x). For every such graph G, it was shown that G ∼ = Γ(P) where P = (V (G) ∪ {0}, ≤) (and 0 ∈ / V (G)) such that, upon fixing a unique representative of v for every v ∈ V (G), the inequality a ≤ b holds in P if and only if either a = 0, a = b, or a, b ∈ V (G) and either cG (b)  cG (a) or a is the unique representative of b. For example, if G is the graph in Fig. 2(a), then this construction yields the poset P1 whose Hasse diagram is given in Fig. 2(b). By introducing an extra condition (namely, the hypothesis in Theorem 4.4(2)) and revising the definition of ≤ in a way such that v has a certain linear ordering for every v ∈ V (G) (as described below), graphs that are realizable as

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zero-divisor graphs of meet-semilattices (equivalently, commutative Boolean semigroups) are characterized in Theorem 4.4. Let G be a simple graph. For every equivalence class v = {u ∈ V (G) | c(v) = c(u)} of vertices in G, fix a linear order on v. Define a relation ≤ on V (G) ∪ {0, 1} (where 0, 1 ∈ / V (G)) by declaring a ≤ b if and only if either a = 0 or b = 1, or a, b ∈ V (G) and either cG (b)  cG (a), or cG (a) = cG (b) and a is less than or equal to b in the linear ordering on a. It is straightforward to check that P(G) := (V (G) ∪ {0, 1}, ≤) is a poset with least element 0 and greatest element 1. For example, if G is the graph in Fig. 2(a), then this construction yields the poset P2 = P(G) whose Hasse diagram is given in Fig. 2(c). Observe that notation is being abused since P(G) depends on the linear orderings of the equivalence classes v. (On the other hand, the zero-divisor graph of P(G) is independent of the linear orderings on these equivalence classes; cf. Lemma 4.3.) By further abuse, if v is endowed with a bounded linear ordering (that is, a linear ordering with a least element and a greatest element) for every v ∈ V (G), then we shall write P  (G) := (V (G) ∪ {0, 1}, ≤). Note that P  (G) always exists (e.g., given any v ∈ G, either v = {v}, or one can endow v \ {v} with a well-ordering and then declare a ≤ v for every a ∈ v). Intuitively, Lemma 4.1 suggests that any set of two adjacent vertices in G has no nonzero lower bound in P(G), and these vertices are therefore adjacent in Γ(P(G)). Similarly, if G is isomorphic to the zerodivisor graph of a poset, then Lemma 4.2 suggests that nonadjacent vertices in G are also nonadjacent in Γ(P(G)). These observations are made formal in the following lemma (cf. the proof of [19, Theorem 3.1]). Lemma 4.3. Let G be a simple connected graph with |V (G)| ≥ 2. Suppose that, for every a, b ∈ V (G) such that a ∈ / cG (b), there exists an x ∈ V (G) such that NG (a, b) ⊆ cG (x). Then G = Γ(P(G)). In particular, ∼ G = Γ(P) for some poset P if and only if G = Γ(P(G)). Proof. Suppose that a, b ∈ V (G) such that a ∈ cG (b). Then NG (a, b)  cG (x) for every x ∈ V (G) by Lemma 4.1. Thus, in P(G), the equality {a, b}∧ = {0} holds by the definition of ≤. Hence, a, b ∈ V (Γ(P(G))) and a ∈ cΓ(P(G)) (b). Since G is a simple connected graph with |V (G)| ≥ 2, every element a ∈ V (G) is adjacent to some b ∈ V (G). Moreover, the above argument shows that adjacent vertices in G are also adjacent vertices in Γ(P(G)). It follows that every vertex of G is also a vertex of Γ(P(G)). Of course, the inclusion V (Γ(P(G))) ⊆ V (G) holds by the definition of P(G). Hence, V (G) = V (Γ(P(G))). Therefore, to prove that G = Γ(P(G)), it only remains to prove that adjacent vertices in Γ(P(G)) are also adjacent in G. Suppose that a and b are adjacent in Γ(P(G)). To the contrary, assume that a ∈ / cG (b). By hypothesis, there exists an x ∈ V (G) such that NG (a, b) ⊆ cG (x). Also, since 0 is the only lower bound of {a, b} in P(G), the definition of ≤ implies that the sets cG (a) and cG (b) are incomparable. In particular, the inclusion NG (a, b) ⊆ cG (x) implies that cG (a) and cG (b) are both properly contained in cG (x). But then x is a nonzero lower bound of {a, b} in P(G), which contradicts that a and b are adjacent in Γ(P(G)). Therefore, a ∈ cG (b). Since zero-divisor graphs of posets are connected with at least two vertices [19, Proposition 2.1 and Theorem 3.1], the “only if” portion of the “in particular” statement now follows by the above argument together with Lemma 4.2. The “if” portion of the “in particular” statement is trivial. 2 The next theorem completely characterizes graphs that are realizable as zero-divisor graphs of meetsemilattices. Theorem 4.4. Let G be a graph. There exists a meet-semilattice S with Z(S) \ {0} = ∅ and G ∼ = Γ(S) if and only if G is simple and connected with |V (G)| ≥ 2 such that, for every a, b ∈ V (G), either a ∈ cG (b), or there exists an x ∈ V (G) such that both of the following statements hold.

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Fig. 3. The Hasse diagram and the zero-divisor graph of a poset P. Note that Γ(P)  Γ(S) for every meet-semilattice S since the nonadjacent vertices a and b of Γ(P) fail condition (2) of Theorem 4.4.

(1) NG (a, b) ⊆ cG (x). (2) If y ∈ V (G) such that NG (a, b) ⊆ cG (y), then cG (x) ⊆ cG (y). Proof. Without loss of generality, suppose that G = Γ(S) for some meet-semilattice S with Z(S) \ {0} = ∅. Note that G is simple by the definition of a zero-divisor graph, and G is connected with |V (G)| ≥ 2 by [19, Proposition 2.1 and Theorem 3.1]. Let a, b ∈ V (G). If a ∈ / cG (b), then x = a ∧ b = 0. Thus, x ∈ V (G), and it is clear that NG (a, b) ⊆ cG (x). Hence, x satisfies (1). To show that x satisfies (2), assume that NG (a, b) ⊆ cG (y) for some y ∈ V (G). Let t ∈ cG (x). If t ∈ NG (a, b), then the desired containment t ∈ cG (y) holds. Thus, assume that t ∈ cG (x) \ NG (a, b). Since t ∧ a = 0, it follows that t ∧ a ∈ V (G). Moreover, (t ∧ a) ∧ b = t ∧ (a ∧ b) = t ∧ x = 0, so t ∧ a ∈ NG (a, b) ⊆ cG (y). Hence, 0 = (t ∧ a) ∧ y = (t ∧ y) ∧ a. Therefore, if t ∧ y = 0, then t ∧ y ∈ cG (a) ⊆ NG (a, b) ⊆ cG (y). But then t ∧ y = t ∧ (y ∧ y) = (t ∧ y) ∧ y = 0, which is a contradiction. Thus, t ∧ y = 0, i.e., t ∈ cG (y), and it follows that cG (x) ⊆ cG (y). It remains to verify the “if” portion of the theorem. By Lemma 4.3, it suffices to prove that P  (G) is a meet-semilattice (note that Z(P  (G))\{0} = V (G) = ∅). Let a, b ∈ P  (G). Clearly a ∧ b = minP  (G) {a, b} if a or b is either 0 or 1. Furthermore, Lemma 4.3 implies that a ∧ b = 0 in the poset P  (G) anytime a ∈ cG (b). Therefore, to show that P  (G) is a meet-semilattice, it only remains to show that a ∧ b exists in P  (G) whenever a, b ∈ V (G) and the hypotheses in (1) and (2) hold for some x ∈ V (G). By the definition of ≤, the elements a and b are comparable in P  (G) anytime cG (a) and cG (b) are comparable under inclusion. In this case, a ∧ b = minP  (G) {a, b}. Therefore, assume that cG (a) and cG (b) are incomparable. Let x ∈ V (G) be an element that satisfies (1) and (2), and assume that x is the maximum element of x. Since cG (a) and cG (b) are incomparable, note that condition (1) implies cG (a) and cG (b) are both properly contained in cG (x ). Hence, x ∈ {a, b}∧ . Also, if y ∈ {a, b}∧ , then the definition of ≤ implies that either y = 0 or NG (a, b) ⊆ cG (y). By the maximality of x together with (2), it follows that y ≤ x . Therefore, a ∧ b = x . 2 Remark 4.5. Let P1 and P2 be the posets whose Hasse diagrams are given in Fig. 2. Then P1 is not a meet-semilattice (since x ∧ y does not exist), but P2 is a meet-semilattice. This shows that it is possible for a poset that is not a meet-semilattice to have a zero-divisor graph which is isomorphic to that of a meet-semilattice. On the other hand, there exist posets whose zero-divisor graphs are not realizable as zero-divisor graphs of meet-semilattices. For example, let P be the poset whose Hasse diagram is given in Fig. 3(a). The zero-divisor graph of P is given in Fig. 3(b). Note that if NΓ(P) (a, b) ⊆ cΓ(P) (x) then x ∈ {x1 , x2 }. But if x = x1 , then condition (2) of Theorem 4.4 fails by setting y = x2 . Symmetrically, setting y = x1 shows that condition (2) of Theorem 4.4 fails if x = x2 . Therefore, Γ(P)  Γ(S) for every meet-semilattice S.

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Fig. 4. The graph G is not the zero-divisor graph of a poset since L(G) is not a Boolean algebra. In particular, although L1 (G) is a lattice, G is not the zero-divisor graph of a meet-semilattice.

The following corollary gives a graph-theoretic characterization of zero-divisor graphs of meet-semilattices in terms of L1 (G) and L(G), and provides a graph-theoretic generalization of the join-semilattice A1 (R) = {ann(r) | r ∈ R} for a reduced commutative ring R with 1 = 0 (cf. Remark 4.8). Corollary 4.6. Let G be a simple connected graph with |V (G)| ≥ 2. Then the following statements are equivalent. (1) (2) (3) (4)

G∼ = Γ(S) for some meet-semilattice S. L1 (G) is a join-semilattice, and L(G) is a Boolean algebra. L1 (G) is a join-semilattice, and G ∼ = Γ(P) for some poset P. L1 (G) is a join-semilattice, and c(a) ∨ c(b) = V (G) for every a, b ∈ V (G) with a ∈ / c(b).

Proof. The equivalence of (2) and (3) holds by [17, Theorem 3.3]. To show that (1) implies (3), note that ∅ ∨ X = X and V (G) ∨ X = V (G) for every X ∈ L1 (G). Also, conditions (1) and (2) of Theorem 4.4 / c(b). Thus, to verify that L1 (G) is a imply that c(a) ∨ c(b) exists in L1 (G) for every a, b ∈ V (G) with a ∈ join-semilattice, it only remains to prove that c(a) ∨ c(b) exists in L1 (G) for every a, b ∈ V (G) with a ∈ c(b). Let a, b ∈ V (G) with a ∈ c(b). By Lemma 4.1, N (a, b)  c(x) for every x ∈ V (G). It follows that c(a) ∨ c(b) = V (G). Hence, L1 (G) is a join-semilattice. Finally, (3) holds by letting P = S since every meet-semilattice is a poset. / c(b). The inclusion N (a, b) ⊆ c(x) holds for To show that (3) implies (4), let a, b ∈ V (G) such that a ∈ some x ∈ V (G) by Lemma 4.2. Thus, c(a) ∨ c(b) = V (G), and hence (4) holds. It only remains to show that (4) implies (1). Suppose that (4) holds. Let a, b ∈ V (G). Then either a ∈ c(b), or c(a) ∨ c(b) = V (G), i.e., either a ∈ c(b), or c(a) ∨ c(b) = c(x) for some x ∈ V (G) (note that c(a) ∨ c(b) = ∅ because G is connected and |V (G)| ≥ 2). That is, either a ∈ c(b), or there exists an x ∈ V (G) satisfying (1) and (2) of Theorem 4.4. Therefore, (1) holds by Theorem 4.4. 2 The next example gives a simple application of Corollary 4.6. Also, it shows that, to conclude that G is the zero-divisor graph of a meet-semilattice, it is not enough to observe that L1 (G) is a join-semilattice. Example 4.7. Let G be the graph in Fig. 4(a). Then L1 (G) is a lattice (see Fig. 4(b)). In fact, it is straightforward to check that L1 (G) = L(G). Note that L(G) is not a Boolean algebra (e.g., complements in L(G) are not unique), and thus G  Γ(S) for every meet-semilattice S by Corollary 4.6.

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Remark 4.8. Let R be a reduced commutative ring with 1 = 0 that contains nonzero zero-divisors. As noted in Section 3, |V (Γ(R))| ≥ 2, L1 (Γ(R)) is a join-semilattice, and L(Γ(R)) is a Boolean algebra. Hence, Corollary 4.6 implies that Γ(R) ∼ = Γ(S) for some meet-semilattice S. In fact, it is observed in [17, Remark 3.4] that (the ring-theoretic zero-divisor graph) Γ(R) is equal to the zero-divisor graph of the poset (R, ≤), where r ≤ s if and only if either annR (s)  annR (r), or annR (r) = annR (s) and r is less than or equal to s in some predetermined linear order on the set [r] = {x ∈ R | annR (x) = annR (r)}. As in the above definition of P  (G), it can be assumed that every such [r] has a least element and a greatest element under its predetermined order. In this case, given any r, s ∈ R, it is straightforward to check that r ∧ s ∈ {r, s} if r and s are comparable, and otherwise r ∧ s is the greatest element of [rs]. Therefore, (R, ≤) is a meet-semilattice whose zero-divisor graph is equal to (the ring-theoretic zero-divisor graph) Γ(R). From these observations, it follows that Corollary 4.6 can be viewed as a generalization from zero-divisor graphs Γ(R) of reduced commutative rings to zero-divisor graphs of meet-semilattices (in particular, the well known result “A1 (R) is a join-semilattice, and A(R) is a Boolean algebra” is recovered from Corollary 4.6). Furthermore, this is a proper generalization; for example, every complete graph can be realized as the zero-divisor graph of a lattice (e.g., consider the lattices S such that a and b are incomparable for every a, b ∈ S \ {0, 1}), but Γ(R) is complete for a reduced commutative ring R if and only if R is a Boolean ring with |R| = 4 [5, Theorem 2.8]. On the other hand, Corollary 1.2 shows that the graphs satisfying the conditions in Corollary 4.6 are precisely the zero-divisor graphs of reduced commutative semigroups. Remark 4.9. The partial order ≤ on R given in Remark 4.8 is a refinement of the well known Abian order ≤A on R, which is defined by r ≤A s if and only if rs = r2 . (The order ≤A was introduced in [1], where direct products of fields were characterized in terms of ≤A .) To show that ≤ refines ≤A , suppose that r ≤A s. Clearly annR (s) ⊆ annR (r) (since R is reduced). If annR (s)  annR (r), then r ≤ s by the definition of ≤. If annR (s) = annR (r), then the equality rs = r2 implies s − r ∈ annR (r) = annR (s). Thus, s2 = sr, i.e., s ≤A r. Hence, r = s. In particular, r ≤ s. Therefore, if r ≤A s, then r ≤ s. 5. Zero-divisor graphs of join-semilattices Corollary 4.6 shows that L1 (G) and L(G) determine when G is isomorphic to the zero-divisor graph of a meet-semilattice. It is natural to ask whether the dual statement of Corollary 4.6 holds: I.e., is the statement “L1 (G) is a meet-semilattice, and L(G) is a Boolean algebra” equivalent to “G ∼ = Γ(S) for some join-semilattice S?” The next result addresses this question. Theorem 5.1. Let G be a simple connected graph with |V (G)| ≥ 2. If L1 (G) is a meet-semilattice, and L(G) is a Boolean algebra, then G ∼ = Γ(S) for some join-semilattice S. Proof. Since L(G) is a Boolean algebra, it follows that G is isomorphic to the zero-divisor graph of a poset [17, Theorem 3.3]. Therefore, G ∼ = Γ(P  (G)) by Lemma 4.3. Hence, it suffices to prove that P  (G) is a join-semilattice. Let a and b be any two elements of the poset P  (G). It is clear that a ∨ b ∈ {a, b} in P  (G) if a or b is either 0 or 1. Similarly, a ∨ b ∈ {a, b} if cG (a) and cG (b) are comparable. Thus, assume that a, b ∈ V (G) and cG (a) and cG (b) are incomparable. If cG (v)  cG (a) ∩cG (b) for every v ∈ V (G), then the definition of ≤ implies that {a, b}∨ = {1} in P  (G), i.e., a∨b = 1. Therefore, assume that cG (v) ⊆ cG (a)∩cG (b) for some v ∈ V (G). It follows that c(a)∧c(b) = ∅. Clearly cG (a) ∧ cG (b) = V (G). Hence, there exists an x ∈ V (G) such that cG (x) = cG (a) ∧ cG (b). Let x be the minimum element of x. Since cG (a) and cG (b) are incomparable, it follows that cG (x ) = cG (x) is properly contained in cG (a), and thus a ≤ x . Similarly, b ≤ x . Finally, if y ∈ V (G) ∪ {0, 1} with a ≤ y and b ≤ y, then either y = 1, or y ∈ V (G) and cG (y) ⊆ cG (a) and cG (y) ⊆ cG (b). Thus, either y = 1,

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Fig. 5. S is a complete lattice, but L1 (Γ(S)) is not a meet-semilattice since cΓ(S) (a1 ) ∧ cΓ(S) (b1 ) does not exist.

or y ∈ V (G) and cG (y) ⊆ cG (a) ∧ cG (b) = cG (x). By the minimality of x , it follows that x ≤ y. Therefore, a ∨ b = x . 2 Let G be a simple connected graph with |V (G)| ≥ 2. If L1 (G) is a join-semilattice and L(G) is a Boolean algebra, then Corollary 4.6 shows that G is isomorphic to the zero-divisor graph of a meet-semilattice. In this case, G satisfies the conditions of Theorem 4.4. By the proof of Theorem 4.4, it follows that P  (G) is a meet-semilattice. Furthermore, the proof of Theorem 5.1 shows that P  (G) is a join-semilattice whenever L1 (G) is a meet-semilattice and L(G) is a Boolean algebra. By setting S = P  (G), Lemma 4.3 yields the following corollary. Corollary 5.2. Let G be a simple connected graph with |V (G)| ≥ 2. If L1 (G) is a lattice, and L(G) is a Boolean algebra, then G ∼ = Γ(S) for some lattice S. Note that, by Example 4.7, the hypothesis “L(G) is a Boolean algebra” in Theorem 5.1 and Corollary 5.2 cannot be dropped. In contrast to Corollary 4.6, the next example shows that the converses of Theorem 5.1 and Corollary 5.2 are false. Example 5.3. Let N be the set of positive integers, and Ei = {2n | i ≤ n ∈ N}. For every i ∈ N, de fine xi = {i}. Also, let ai = x2 ∪ Ei+2 and bi = x4 ∪ Ei+2 . Finally, let ci = N \ {1, 3, 5, . . . , 2i − 1} ∪  {6, 8, 10, . . . , 4 + 2i} . Then S = ({∅, N} ∪ {x2i−1 , ai , bi , ci }∞ i=1 , ⊆) is a complete lattice. However, L1 (Γ(S)) is not a meet-semilattice. Proof. It is not difficult to check that S is a complete lattice. (For this, it is helpful to note that, for every ∅ = ∞ ∞ A ⊆ S, either A ⊆ {ai , ci }∞ i=1 , A ⊆ {bi , ci }i=1 , A ⊆ {xj } ∪ {ci }i=1 for some j ∈ N, or inf S (A) = ∅; see Fig. 5. Also, see the comments prior to Lemma 5.4.) However, note that cΓ(S) (a1 ) ∩ cΓ(S) (b1 ) = {x1 , x3 , x5 , . . .}, and hence cΓ(S) (t) ⊆ cΓ(S) (a1 ) ∩ cΓ(S) (b1 ) if and only if t ∈ {c1 , c2 , c3 , . . .}. Since cΓ(S) (c1 )  cΓ(S) (c2 )  cΓ(S) (c3 )  · · · is strictly increasing and nonterminating, it follows that cΓ(S) (a1 ) ∧ cΓ(S) (b1 ) does not exist in L1 (Γ(S)). 2 Let G be a simple graph. Of course, if V (G) is finite, then L1 (G) is also finite (it is a subset of the power set of V (G)). In particular, if P is a finite poset with 0, then L1 (Γ(P)) is finite. On the other hand, it can happen that L1 (G) is finite even if G is infinite. For example, if G is any (possibly infinite) complete bipartite graph, then L(G) = L1 (G) is isomorphic to the Boolean algebra on four elements. More generally,

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the mapping given by v → c(v) from V (G) into L1 (G) \ {∅, V (G)} is bijective, and therefore L1 (G) is finite if and only if V (G) is finite. Similarly, it can happen that Γ(P) is finite even if P is infinite (e.g., introducing an infinite chain of upper bounds of Z(P) into P will not affect Γ(P)). However, it is straightforward to check that the subposet Z(P) = V (Γ(P)) ∪ {0} of any poset P satisfies the equality Γ(P) = Γ(Z(P)). Furthermore, if P is a meet-semilattice (respectively, join-semilattice), then Z(P) ∪ {1} is also a meet-semilattice (respectively, join-semilattice) and Γ(P) = Γ(Z(P) ∪{1}). Therefore, the zero-divisor graph of any poset, meet-semilattice, or join-semilattice is finite if and only if it is the zero-divisor graph of a finite poset, finite meet-semilattice, or finite join-semilattice, respectively. Recall that a bounded meet-semilattice (respectively, join-semilattice) is complete if and only if it is a complete lattice (e.g., sup(A) = inf(A∨ ) for any nonempty subset A of a complete meet-semilattice). In particular, every finite bounded semilattice is a lattice. We summarize this discussion in the following lemma. Lemma 5.4. Let G be a simple graph. Then the following statements hold. (1) L1 (G) is finite if and only if V (G) is finite. (2) V (G) is finite and G ∼ = Γ(Q) for a poset Q if and only if G ∼ = Γ(P) for some finite poset P. (3) V (G) is finite and G ∼ = Γ(Q) for a meet-semilattice (respectively, join-semilattice) Q if and only if G∼ = Γ(S) for some finite lattice S. We are prepared to prove Corollary 1.1, which shows that the converses of Theorem 5.1 and Corollary 5.2 hold for finite graphs. Proof of Corollary 1.1. The assumption |V (G)| < ∞ implies that L1 (G) is finite, and thus (1), (2), and (3) are equivalent by the above discussion. Similarly, (4), (5) and (6) are equivalent by Lemma 5.4(3). Finally, (3) and (5) are equivalent by Corollary 4.6. 2 Remark 5.5. For a simple connected graph G with 2 ≤ |V (G)| < ∞, Lemma 5.4(3) implies that the conditions in Corollary 1.1 are equivalent to the condition “G ∼ = Γ(S) for some finite lattice S.” It is natural to ask whether the conditions in Corollary 1.1 hold for infinite graphs if the “S is finite” assumption is relaxed to “S is complete.” Note that Example 5.3 answers this question in the negative. In particular, if V (G) is infinite and (4), (5), and (6) of Corollary 1.1 hold for a complete lattice S, then the join-semilattice L1 (G) (see Corollary 4.6) need not be complete. Remark 5.6. Let R be a reduced commutative ring with 1 = 0 that contains nonzero zero-divisors. As noted in Section 3, |V (Γ(R))| ≥ 2, L1 (Γ(R)) is a join-semilattice, and L(Γ(R)) is a Boolean algebra. If L1 (Γ(R)) is a lattice, then A1 (R) = {annR (r) | r ∈ R} is a lattice (see Section 3), and Γ(R) is isomorphic to the zero-divisor graph of a lattice by Corollary 5.2. In fact, it is not difficult to check that the meet-semilattice (R, ≤) defined in Remark 4.8, whose zero-divisor graph is equal to (the ring-theoretic zero-divisor graph) Γ(R), is a lattice such that r ∨ s ∈ {r, s} if r and s are comparable, and otherwise r ∨ s is the least element of [x], where annR (x) = annR (r) ∧ annR (s). 6. Zero-divisor graphs of reduced commutative semigroups As noted earlier, the zero-divisor graph concept was first applied to general commutative semigroups in [11]. Several articles that examine realizability of zero-divisor graphs of semigroups have since appeared (e.g., see [9,10,19]). For example, condition (1) of Theorem 4.4, which is used to characterize zero-divisor

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graphs of posets in [19] and zero-divisor graphs of meet-semilattices in the present paper, originates in [9, Theorem 1(4)], where it was proved that if x and y are nonadjacent vertices in the zero-divisor graph of a (not necessarily reduced) commutative semigroup, then there exists a vertex z such that N (x, y) ⊆ c(z)∪{z}. However, in [9, Example 2], a graph which satisfies this condition but is not the zero-divisor graph of any commutative semigroup is provided. For a simple graph G that has a finite clique number, it is determined in [19, Theorem 4.3] (based purely on graph-theoretic information) when G ∼ = Γ(S) for some reduced commutative semigroup S. In this section, the assumptions on the clique number are dropped, providing a complete characterization of graphs that are realizable as zero-divisor graphs of reduced commutative semigroups. In [3, Theorem 2.4], it is shown that if R is a reduced commutative ring with 1 = 0, then Γ(RE ) ∼ = Γ(R). The following result extends [3, Theorem 2.4] to reduced commutative semigroups. Theorem 6.1. Let S be a reduced commutative semigroup. Then Γ(SE ) ∼ = Γ(S). Proof. Consider the mapping from V (Γ(SE )) to V (Γ(S)) defined by [a] → a. For a general semigroup S, it can happen that [a] = [0] for some 0 = a ∈ S. However, the hypotheses of the present theorem guarantee that S does not have any nonzero nilpotent elements. It follows that the containment [a] ∈ V (Γ(SE )) holds if and only if a ∈ V (Γ(S)). Clearly the containment a ∈ V (Γ(S)) holds if and only if a ∈ V (Γ(S)). Therefore, given any a ∈ S, the containment [a] ∈ V (Γ(SE )) holds if and only if a ∈ V (Γ(S)). Also, the equality [a] = [b] holds in SE if and only if annS (a) \ {0} = annS (b) \ {0}. If a, b ∈ V (Γ(S)), then the last equality is equivalent to cΓ(S) (a) = cΓ(S) (b) (since S does not have any nonzero nilpotent elements). Thus, [a] = [b] in V (Γ(SE )) if and only if a = b in V (Γ(S)). It follows that the above mapping is a well-defined bijection. Moreover, since S does not have any nonzero nilpotent elements, the relations [a] ∈ cΓ(SE ) ([b]) and a ∈ cΓ(S) (b) are both equivalent to the containment a ∈ cΓ(S) (b). Therefore, the above mapping preserves and reflects all adjacency relations. 2 Remark 6.2. Note that Theorem 6.1 can fail if the “reduced” hypothesis is dropped. For example, consider the usual (multiplicative) semigroup S = Z9 of integers modulo 9. Then Γ(SE ) ∼ = K1  K2 ∼ = Γ(S), where Kn denotes the complete graph on n vertices. Let G be a simple connected graph such that G ∼ = Γ(Q) for some commutative Boolean semigroup Q. In [19, Proposition 4.2], it was shown that a Boolean semigroup S can be constructed such that G ∼ = Γ(S). By using Theorem 4.4, we are able to give an alternative proof of this fact; in particular, the proof of the next lemma avoids the tedious verification of associativity in S. Lemma 6.3. Let G be a simple connected graph such that G ∼ = Γ(Q) for some commutative Boolean semi∼ group Q. Then G = Γ(S) for some Boolean semigroup S. Proof. The zero-divisor graph of any Boolean semigroup Q is isomorphic to the zero-divisor graph of a meet-semilattice (namely, it is the zero-divisor graph of the meet-semilattice corresponding to Q; see Section 2). Hence, G satisfies conditions (1) and (2) of Theorem 4.4. By the definition of G, the surjective mapping ϕ : V (G) → V (G) defined by ϕ(a) = a preserves and reflects adjacency. It follows that G also satisfies conditions (1) and (2) of Theorem 4.4. Therefore, G is the zero-divisor graph of a meet-semilattice. The result follows since the zero-divisor graph of any meet-semilattice S is the zero-divisor graph of the Boolean semigroup corresponding to S. 2 We are prepared to prove Corollary 1.2, which completely characterizes graphs that are realizable as zero-divisor graphs of reduced commutative semigroups.

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Proof of Corollary 1.2. Note that (2) implies (1) and the equivalence of (2) and (3) hold trivially. Also, (3), (4), (5), (6) and (7) are equivalent by Theorem 4.4 and Corollary 4.6. It remains to verify that (1) implies (2). Assume that (1) holds. Note that SE is a commutative Boolean semigroup since S is reduced. Also, G ∼ = Γ(S) ∼ = Γ(SE ) by Theorem 6.1. By Lemma 6.3, G is isomorphic to the zero-divisor graph of a commutative Boolean semigroup. Thus, (2) holds. 2 References [1] A. Abian, Direct product decomposition of commutative semisimple rings, Proc. Am. Math. Soc. 24 (1970) 502–507. [2] D.F. Anderson, M.C. Axtell, J.A. Stickles Jr., Zero-divisor graphs in commutative rings, in: M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson (Eds.), Commutative Algebra, Noetherian and Non-Noetherian Perspectives, Springer-Verlag, New York, 2011, pp. 23–45. [3] D.F. Anderson, J.D. LaGrange, Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra 216 (2012) 1626–1636. [4] D.F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003) 221–241. [5] D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434–447. [6] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226. [7] B. Bollobás, Modern Graph Theory, Springer, New York, 1998. [8] J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, S. Spiroff, On zero divisor graphs, in: C. Francisco, et al. (Eds.), Progress in Commutative Algebra II: Closures, Finiteness and Factorization, de Gruyter, Berlin, 2012. [9] F. DeMeyer, L. DeMeyer, Zero-divisor graphs of semigroups, J. Algebra 283 (2005) 190–198. [10] L. DeMeyer, Y. Jiang, C. Loszewski, E. Purdy, Classification of zero-divisor semigroup graphs, Rocky Mt. J. Math. 40 (2010) 1481–1503. [11] F. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002) 206–214. [12] E. Estaji, K. Khashyarmanesh, The zero-divisor graph of a lattice, Results Math. 61 (2012) 1–11. [13] G. Grätzer, General Lattice Theory, Birkhäuser Verlag, Basel, 1978. [14] R. Halaš, M. Jukl, On Beck’s coloring of posets, Discrete Math. 309 (2009) 4584–4589. [15] R. Halaš, H. Länger, The zerodivisor graph of a qoset, Order 27 (2010) 343–351. [16] J.D. LaGrange, On realizing zero-divisor graphs, Commun. Algebra 36 (2008) 4509–4520. [17] J.D. LaGrange, K.A. Roy, Poset graphs and the lattice of graph annihilators, Discrete Math. 313 (10) (2013) 1053–1062. [18] J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966. [19] D. Lu, T. Wu, The zero-divisor graphs of posets and an application to semigroups, Graphs Comb. 26 (2010) 793–804. [20] S.B. Mulay, Cycles and symmetries of zero-divisors, Commun. Algebra 30 (2002) 3533–3558. [21] S. Spiroff, C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Commun. Algebra 39 (2011) 2338–2348.