# On Reduced Zero-Divisor Graphs of Posets

Authors: Ashish Kumar Das; Deiborlang Nongsiang
Journal: Journal of Discrete Mathematics
 (2015)

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## Abstract

We study some properties of a graph which is constructed from the equivalence classes of nonzero zero-divisors determined by the annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.


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## Body

## 1. Introduction

The study of the interrelationship between algebra and graph theory by associating a graph to an algebraic object was initiated, in 1988, by Beck [1] who developed the notion of a zero-divisor graph of a commutative ring with identity. Since then, a number of authors have studied various forms of zero-divisor graphs associated to rings and other algebraic structures (see, e.g., [2–5]).In 2009, Halaš and Jukl [6] introduced the notion of a zero-divisor graph of a partially ordered set (in short, a poset). The study of various types of zero-divisor graphs of posets was then carried out by many others in [7–9]. However, the zero-divisor graph of a poset considered in this paper was actually introduced by Lu and Wu [10], which is slightly different from the one introduced in [6].In this paper, inspired by the ideas of Mulay [11] and Spiroff and Wickham [12], we study some properties of a graph which is constructed from the equivalence classes of nonzero zero-divisors determined by the annihilator ideals of a poset. This graph is the same as the reduced graph of the zero-divisor graph of a poset (see [10, page 798]). In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.


## 2. Prerequisites

In this section, we put together some well-known concepts, most of which can be found in [13–16].We begin by recalling some of the basic terminologies from the theory of graphs. Needless to mention that all graphs considered here are simple graphs, that is, without loops or multiple edges. LetG be a graph and x,y∈V(G) the vertex set of G. Then, x and y are said to beadjacent if x≠y and there is an edge x-y between x and y. Awalk between x and y is a sequence of adjacent vertices, often written as x-x1-x2-⋯-xn-y. A walk between x and y is calledpath if the vertices in it are all distinct (except, possibly, x and y). A path between x and y is called acycle if x=y. The number of edges in a path or a cycle is called itslength. If x≠y, then the minimum of the lengths of all paths between x and y in G is called thedistance between x and y in G and is denoted by dist⁡(x,y). If there is no path between x and y, then we define dist⁡(x,y)=∞. The maximum of all possible distances in G is called thediameter of G and is denoted by diam⁡(G). The girth of a graph G is the minimum of the lengths of all cycles in G and is denoted by girth(G). If G isacyclic, that is, if G has no cycles, then we write girth(G)=∞. Acycle graph is a graph that consists of a single cycle (an n-gon).A subset of the vertex set of a graphG is called aclique of G if it consists entirely of pairwise adjacent vertices. The least upper bound of the sizes of all the cliques of G is called theclique number of G and is denoted by ω(G).Theneighborhood of a vertex x in a graph G, denoted by nbd(x), is defined to be the set of all vertices adjacent to x while thedegree of x in G, denoted by deg⁡(x), is defined to be the number of vertices adjacent to x, and so deg⁡(x)=nbd(x). If deg⁡(x)=1, then x is said to be anend vertex in G. If deg⁡(x)=deg⁡(y) for all x,y∈V(G), then the graph G is said to be aregular graph.A graphG is said to beconnected if there is a path between every pair of distinct vertices in G. A graph G is said to becomplete if there is an edge between every pair of distinct vertices in G. We denote the complete graph with n vertices by Kn. An r-partite graph, r≥2, is a graph whose vertex set can be partitioned into r disjoint parts in such a way that no two adjacent vertices lie in the same part. Among the r-partite graphs, thecompleter-partite graph is the one in which two vertices are adjacent if and only if they lie in different parts. The complete r-partite graph with parts of size n1,n2,…,nr is denoted by Kn1,n2,…,nr. The 2-partite and complete 2-partite graphs are popularly known asbipartite andcomplete bipartite graphs, respectively. A bipartite graph of the form K1,n is also known as astar graph.Next we turn to partially ordered sets and their zero-divisor graphs. A nonempty set is said to be apartially ordered set (in short, aposet) if it is equipped with apartial order, that is, a reflexive, antisymmetric, and transitive binary relation. It is customary to denote a partial order by “≤.”LetQ be a nonempty subset of a poset P. If there exists y∈Q such that y≤x for every x∈Q, then y is called theleast element of Q. The least element of P, if exists, is usually denoted by 0. An element x∈Q is called aminimal element of Q if y∈Q and y≤x imply that y=x. We denote the set of minimal elements of Q by Min⁡(Q).LetP be a poset with least element 0. An element x∈P is called azero-divisor of P if there exists y∈P×∶=P∖{0} such that the set L(x,y):={z∈P∣z≤xandz≤y}={0}. We denote the set of zero-divisors of P by Z(P) and write Z(P)×∶=Z(P)∖{0}. By anideal of P we mean a nonempty subset I of P such that y∈I whenever y≤x for some x∈I. We say that the ideal I isproper if I≠P. For each x∈P, it is easy to see that the set (x]∶={y∈P∣y≤x} is an ideal of P, called theprincipal ideal of P generated by x. Given x∈P, theannihilator of x in P is defined to be the set ann(x)∶={y∈P∣L(x,y)={0}}, which is also an ideal of P. Note that x∉ann(x) for all x∈P×. A proper ideal p of P is called aprime ideal of P if, for every x,y∈P, L(x,y)⊆p implies that either x∈p or y∈p. A prime ideal p of P is said to be anannihilator prime ideal (or, anassociated prime) if there exists x∈P such that p=ann(x). Two annihilator prime ideals ann(x) and ann(y) are distinct if and only if L(x,y)={0} (see [6, Lemma 2.3]). We write Ann(P) to denote the set of all annihilator prime ideals of P.LetP be a poset with least element 0 and with Z(P)×≠∅. As in [10], thezero-divisor graph of P is defined to be the graph Γ(P) in which the vertex set is Z(P)×, and two vertices x and y are adjacent if and only if L(x,y)={0}. Clearly, Γ(P) is a simple graph; in fact, nbd(x)=ann(x)∖{0} for all x∈V(Γ(P)). It is well known that Γ(P) is a connected graph with diam⁡(Γ(P))∈{1,2,3} and girth(Γ(P))∈{3,4,∞} (see, e.g., [9]). The clique number ω(Γ(P)) of Γ(P) is usually denoted, in short, by ω(P). Clearly, ω(P)≥2. It may be noted here that ω(P)=clique(P)-1, where clique(P) is the clique number of the zero-divisor graph considered in [6] whose vertex set is the whole of P.Finally, we would like to mention that all posets considered in this paper are with least element0 and have nonzero zero-divisors, unless explicitly written otherwise.


## 3. Reduced Graph ofΓ(P): Basic Properties

In this section, we study some of the basic properties of the reduced graph of the zero-divisor graph of a poset.LetP be a poset with least element 0 and with Z(P)×≠∅. Given x,y∈P, set x~y if ann(x)=ann(y). Clearly, ~ is an equivalence relation in P. Let [x] denote the equivalence class of x∈P. Note that if x∈Z(P)×, then [x]⊆Z(P)×. Also, note that [0]={0} and [x]=P∖Z(P) for all x∈P∖Z(P). In analogy with [11, 12], thegraph of equivalence classes of zero-divisors of P may be defined to be the graph ΓE(P) in which the vertex set is the set of all equivalence classes of the elements of Z(P)×, and two vertices [x] and [y] are adjacent if and only if L(x,y)={0}, that is, if and only if x and y are adjacent in Γ(P). Note that two adjacent vertices in Γ(P) represent two distinct equivalence classes and, hence, two distinct vertices in ΓE(P). Thus ΓE(P) is also a simple graph. It may be recalled (see [12]) that, in case of a commutative ring R with unity, two adjacent vertices in Γ(R) do not necessarily represent two distinct vertices in ΓE(R). Since nbd(x)=ann(x)∖{0} for all x∈V(Γ(P)), one obtains the same graph ΓE(P) if the equivalence relation ~ is defined on V(Γ(P))=Z(P)× by setting x~y, where x,y∈V(Γ(P))=Z(P)×, if and only if nbd(x)=nbd(y) in Γ(P). Thus, the graph ΓE(P) is the same as the reduced graph of Γ(P) considered by Lu and Wu in [10]. In particular, ΓE(P) is also a zero-divisor graph of some poset, and, hence, there are lots of structural similarities between Γ(P) and ΓE(P). There are however some more features of ΓE(P) which are worth looking into.The graphΓE(P) has some advantages over the zero-divisor graph Γ(P). In many cases ΓE(P) is finite when Γ(P) is infinite. For example, consider the poset P0={∅,{1},{2},{2,3},{2,4},…} under the set inclusion with least element ∅. Then Γ(P0) is infinite, whereas ΓE(P0) is finite and has only two vertices. In fact, the zero-divisors {2},{2,3},{2,4},…, have the same annihilator and so they represent a single vertex in ΓE(P0); the other vertex in ΓE(P0) is represented by {1}. Another important aspect of ΓE(P) is its connection to the annihilator prime ideals of the poset P, which we discuss in detail in the next section. Let us now rewrite a fact, just noted above, in a more explicit manner for the ease of its extensive use (often without a mention) in this paper.Fact 1.
Given a posetP, two vertices in Γ(P) are adjacent if and only if they represent two adjacent vertices in ΓE(P).
In view of Fact1, it is easy to see that, given a poset P, we have ω(P)=ω(ΓE(P)); that is, the clique numbers of Γ(P) and ΓE(P) are the same.
By [10, Corollary 3.3 (2)], we know that ΓE(P) is also a zero-divisor graph of some poset. Therefore, in view of [9, Theorem 3.3], ΓE(P) is a connected graph with diam⁡ΓE(P)≤3. Our first result of this section not only generalizes Fact 1 but also shows some similarity between Γ(P) and ΓE(P) as far as their diameter is concerned.Proposition 1.
LetP be a poset. If [x] and [y] are two distinct vertices in ΓE(P), then dist⁡([x],[y])=dist⁡(x,y). Moreover, the following assertions hold: (a)
diam⁡(ΓE(P))=3 if and only if diam⁡(Γ(P))=3,(b)
diam⁡(ΓE(P))=2 if and only if diam⁡(Γ(P))=2 and ann(x)≠ann(y) for some x,y∈Z(P)× with L(x,y)≠{0},(c)
diam⁡(ΓE(P))=1 if and only if ann(x)=ann(y) for all x,y∈Z(P)× with L(x,y)≠{0}.Proof.
Let[x] and [y] be two distinct vertices in ΓE(P). If [x]-[x1]-⋯-[xn]-[y] is a path in ΓE(P), then, by Fact 1, x-x1-⋯-xn-y is a path in Γ(P). Therefore, dist⁡([x],[y])≥dist⁡(x,y). On the other hand, if x-x1-⋯-xn-y is a path in Γ(P), then, once again using Fact 1, [x]-[x1]-⋯-[xn]-[y] is a walk in ΓE(P). It follows that dist⁡([x],[y])≤dist⁡(x,y), and, hence, the equality holds.
For proving the given assertions, we first note that there exist two distinct vertices[x] and [y] of ΓE(P) such that dist⁡([x],[y])=diam⁡(ΓE(P)). Therefore, it follows from the first half of this proposition that we always have diam⁡(ΓE(P))≤diam⁡(Γ(P)). However, for the reverse inequality it is not enough to have two distinct vertices x,y∈V(Γ(P)) such that dist⁡(x,y)=diam⁡(Γ(P)); we must also have an additional requirement; namely, ann(x)≠ann(y). If dist⁡(x,y)=3, then this additional requirement is guaranteed by the existence of a path of the form x-a-b-y in Γ(P). Hence, the assertion (a) follows. This in turn also proves the assertion (b), because the extra condition included in (b) fulfills the additional requirement mentioned above. The last assertion, namely, (c), now follows from the assertions (a) and (b).From [9, 10], we know that girth(ΓE(P))∈{3,4,∞}. In this context, we have the following result.Proposition 2.
LetP be a poset. Then,
(1)girthΓEP=3⟺girthΓP=3⟺VΓEP≥3.
Consequently, girth(ΓE(P))=∞ if and only if V(ΓE(P))=2.Proof.
Assume thatV(ΓE(P))≥3. Then we also have |V(Γ(P))|≥3. Moreover, Γ(P) is not a star graph; otherwise we would have V(ΓE(P))=2. By [9, Theorem 4.2], we have girth(Γ(P))∈{3,4,∞}. If girth (Γ(P))=4 or ∞, then it follows from [9, Remark 4.12] that Γ(P) is a complete bipartite graph, which in turn implies that V(ΓE(P))=2. Therefore, we have girth (Γ(P))=3. Hence, in view of Fact 1, we have girth (ΓE(P))=3. On the other hand, if girth(ΓE(P))=3 then we obviously have V(ΓE(P))≥3.
The consequential statement is clear as, by the choice ofP, we always have |V(ΓE(P))|≥2.As an immediate consequence, we have the following corollary which may be compared with [12, Proposition 1.8].Corollary 3.
There is no posetP for which ΓE(P) is a cycle graph with at least four vertices.Our next result is a small observation, which has some interesting consequences in contrast to some results of a similar nature in [12].Proposition 4.
LetP be a poset. Then no two distinct vertices of ΓE(P) have the same neighborhood. Equivalently, given x,[y]∈V(ΓE(P)), one has nbd([x])=nbd([y]) if and only if ann(x)=ann(y).Proof.
Let[x] and [y] be two distinct vertices in ΓE(P) such that nbd([x])=nbd([y]). Let z∈ann(x), z≠0. Then, by Fact 1, we have [z]∈nbd([x])=nbd([y]), whence z∈ann(y). Thus, it follows that ann(x)⊆ann(y). Similarly, we have ann(y)⊆ann(x), and so ann(x)=ann(y). This contradiction proves the proposition.The above proposition says, in other words, that the reduced zero-divisor graph of a poset cannot be further reduced (symbolically,ΓE(P)≅(ΓE)E(P)).Corollary 5.
LetP be a poset and r≥2. Then, ΓE(P) is a complete r-partite graph if and only if it is a complete graph with r-vertices.Proof.
LetΓE(P) be a complete r-partite graph with VΓEP=V1⊔V2⊔⋯⊔Vr. Let x,[y]∈Vj, where 1≤j≤r. Then, nbd([x])=nbd([y])=V(ΓE(P))∖Vj, and so, by Proposition 4, [x]=[y]. Thus, Vj=1 for all j∈{1,2,…,r}, which means that ΓE(P) is a complete graph with r vertices. The converse is trivial.Corollary 6.
LetP be a poset such that V(ΓE(P))≥3. Then ΓE(P) is not a star graph.Proof.
It is enough to note that in a star graph with at least three vertices all the end vertices have the same neighborhood. Alternatively, one may also note that star graphs are complete bipartite graphs.Corollary 7.
There is only one graph with exactly three vertices that can be realized as the graphΓE(P) for some poset P, and it is the cycle/complete graph K3.Proof.
Since, given a posetP, the graph ΓE(P) is a connected but not a star graph, we need only to note that the graph of equivalence classes of zero-divisors of the poset {∅,{1},{2},{3}} is precisely K3.One of the obvious consequences of Fact1 is that if for some poset P the graph Γ(P) is complete, then the corresponding graph ΓE(P) is also complete; in fact, ΓE(P)≅Γ(P). Also, it follows from the definition of ΓE(P) that if Γ(P) is a complete r-partite graph, r≥2, then ΓE(P) is a complete graph with r vertices. Thus, for every n≥2, the complete graph Kn can be realized as ΓE(P) for some poset P; for example, we may consider the poset {∅,{1},{2},…,{n}}. This is not the case for rings (see [12, Proposition 1.5]).


## 4. Posets with an Ascending Chain Condition

In this section, imposing certain restrictions on a given poset, we study some properties of its reduced zero-divisor graph in terms of the annihilator prime ideals.LetP be a poset. We say that P isa poset with ACC for annihilators if the ascending chain condition holds for its annihilator ideals, that is, if there is no infinite strictly ascending chain in the set A:={ann(x)∣x∈P×} under set inclusion. Equivalently, P is a poset with ACC for annihilators if and only if every nonempty subset of A has a maximal element. Thus, if P is a poset with ACC for annihilators, then A has a maximal element and every element of A is contained in a maximal element of A. We denote the set of all maximal elements of A by Max⁡(A).IfP is a poset such that ω(P)<∞, then from [6, Lemma 2.4] it follows that P is a poset with ACC for annihilators. In particular, if P is a poset such that deg⁡(x)<∞ for all x∈V(Γ(P)), then P is a poset with ACC for annihilators; noting that Γ(P) has no infinite clique and so, by [6, Lemma 2.10], we have ω(P)<∞. On the other hand, consider the poset
(2)P=∅∪n∣n∈N∪m,m+1,…∣m∈N,m≠1
under set inclusion, where N denotes the set of all positive integers. It is easy to see that there is an infinite strictly ascending chain of annihilator ideals of P given by
(3)ann2,3,…⊊ann3,4,…⊊…,
which means that P is a poset without ACC for annihilators.Remark 8.
IfP is a poset with ACC for annihilators, then the clique number of Γ(P) need not be finite; that is, Γ(P) may contain an infinite clique; for example, consider the poset P={∅}∪{{n}∣n∈N} under set inclusion. This is contrary to what has been asserted in Proposition 2.6 of [10]. In fact, a careful look at the proof of [10, Proposition 2.6] reveals that while proving “(2)⇒(3)” the authors mistakenly assumed the validity of the first statement of the said proposition.Our first result concerning the set of all annihilator prime ideals of a poset is given as follows.Proposition 9.
LetP be a poset. Then,
(4)AnnP=Max⁡A.
In particular, if P is a poset with ACC for annihilators, then Ann(P)≠∅.Proof.
By [6, Lemma 2.2], we have Max⁡(A)⊆Ann(P). Conversely, suppose that ann(x)∈Ann(P)∖Max⁡(A). Then there exists ann(y)∈A such that ann(x)⊊ann(y). Choose z∈ann(y)∖ann(x). Since L(z,y)={0}⊆ann(x) and ann(x) is a prime ideal, it follows that y∈ann(x)⊂ann(y) which is absurd. Hence, we have Ann(P)=Max⁡(A). The particular case follows from the fact that if P is a poset with ACC for annihilators, then Max⁡(A)≠∅.Given a posetP, consider the set B:={ann(x)∣x∈Z(P)×}. Clearly, ∅≠B⊆A; in fact, B=A∖{0}. Since Z(P)×≠∅, {0} is not a prime ideal of P, and so, it follows that Ann(P)⊆B. Note that there is a natural bijective map from B to the vertex set of ΓE(P) given by ann(x)↦[x]. As such, we may treat B as the vertex set of ΓE(P). In view of this, with a slight abuse of terminology, we sometimes refer to [x]∈V(ΓE(P)) as an annihilator ideal (resp., an annihilator prime ideal) if we have ann(x)∈B (resp., ann(x)∈Ann(P)). All the forthcoming results of this section are under this identification.The following lemma plays a crucial role in this section.Lemma 10.
LetP be a poset. Then the following assertions hold. (a)
Givenann(x),ann(y)∈B, one has ann(x)⊊ann(y) if and only if nbd([x])⊊nbd([y]) in ΓE(P).(b)
Givenann(x)∈B and ann(z)∈Ann(P), one has ann(x)⊈ann(z) if and only if L(x,z)={0}; equivalently, one has nbd([x])⊆nbd([z]) if and only if the vertices [x] and [z] are not adjacent in ΓE(P).(c)
Givenann(x),ann(y)∈B, if ann(x)∪ann(y)⊆ann(z) for some ann(z)∈Ann(P), then the vertices [x] and [y] are not adjacent in ΓE(P); that is, L(x,y)≠{0}. The converse is also true if P is a poset with ACC for annihilators.(d)
IfP is a poset with ACC for annihilators, then for each ann(x)∈B there exists ann(z)∈Ann(P) such that ann(x)⊈ann(z).(e)
Ifann(x)∈B∖Ann(P), then there exist u,v∉ann(x) such that L(u,v)={0}.Proof.
(a) In view of Proposition4 and the definition of ΓE(P), it is enough to note that, given u,x∈Z(P)×, one has u∈ann(x) if and only if [u]∈nbd([x]) in ΓE(P).
(b) Ifann(x)⊈ann(z), then, choosing y∈ann(x)∖ann(z), we have L(x,y)={0}⊆ann(z), and so x∈ann(z), since ann(z) is a prime ideal. Conversely, if L(x,z)={0}, then z∈ann(x), and so ann(x)⊈ann(z), since z∉ann(z). The equivalent assertion follows from part (a).
(c) Ifann(x)∪ann(y)⊆ann(z) for some ann(z)∈Ann(P), then, by part (b), we have x,y∉ann(z), which implies that L(x,y)≠{0}, since ann(z) is a prime ideal. Conversely, if L(x,y)≠{0}, then, choosing w∈L(x,y)∖{0}, we have ann(x)∪ann(y)⊆ann(w)⊆ann(z) for some ann(z)∈Ann(P).
(d) Lety∈ann(x)∖{0}. Choose ann(z)∈Ann(P) such that ann(y)⊆ann(z). Since x∈ann(y), we have x∈ann(z), or, equivalently, z∈ann(x). Thus ann(x)⊈ann(z), since z∉ann(z).
(e) Ifann(x)∈B∖Ann(P), then, by Proposition 9, there exists ann(u)∈A such that ann(x)⊊ann(u), and so we may choose v∈ann(u)∖ann(x) to complete the proof.Proposition 11.
LetP be a poset. Then, the following assertions hold. (a)
Ann(P) is a clique of ΓE(P).(b)
Given[z]∈B, one has [z]∈Ann(P) if and only if no two vertices in the set V(ΓE(P))∖nbd([z]) are adjacent.(c)
IfP is a poset with ACC for annihilators, then every vertex in ΓE(P) is adjacent to an annihilator prime ideal; consequently, Ann(P)≥2.Proof.
In view of Fact1, the results follow from Lemma 10. More precisely, the first part follows from part (b) using maximality of the annihilator prime ideals, the second part from parts (b), (c), and (e), and the third part from parts (b) and (d) of Lemma 10. The first part also follows directly from [6, Lemma 2.3].IfP is a poset with ω(P)<∞, then, in view of [6, Lemmas 2.6], it follows from Proposition 9 that Ann(P)<∞. In this context, we have a stronger result in the following form.Proposition 12.
LetP be a poset with ACC for annihilators. Then,
(5)AnnP=ωP.
In particular, Ann(P)≥3 if and only if V(ΓE(P))≥3.Proof.
By Proposition11(a), we have Ann(P)≤ω(ΓE(P))=ω(P). Conversely, suppose that {x1,x2,…,xk} is a clique of Γ(P). For each i∈{1,2,…,k}, choose ann(zi)∈Max⁡(A)=Ann(P) such that ann(xi)⊆ann(zi). By Lemma 10(c), the annihilator prime ideals ann(z1),ann(z2),…,ann(zk) are all pairwise distinct. It follows that Ann(P) is an upper bound for the sizes of all the cliques of Γ(P), and hence, Ann(P)≥ω(P). This proves the first part. The particular case follows from Proposition 2.It follows immediately from Proposition11(a) and Proposition 12 that, given a poset P with ACC for annihilators, Ann(P) is a clique of maximal size in ΓE(P).Remark 13.
LetP be a poset with ACC for annihilators. Then, as a trivial consequence of Proposition 12, we have V(ΓE(P))≥3 if and only if no annihilator prime ideal in ΓE(P) is an end vertex, and so, by Proposition 11(c), every vertex in ΓE(P) that is adjacent to an end vertex is an annihilator prime ideal (compare with [12, Proposition 3.2 and Corollary 3.3]).Given a posetP, if the degree of some annihilator prime ideal in ΓE(P) is infinity, then obviously V(ΓE(P))=∞. We have, however, a stronger converse given by the following result which may be compared and contrasted with [12, Proposition 2.2].Proposition 14.
LetP be a poset. If the vertex set of ΓE(P) is infinite, then the degree of each annihilator prime ideal in ΓE(P) is infinity.Proof.
Suppose thatV(ΓE(P))=∞. In view of Proposition 11(a), we may assume that Ann(P)<∞. In that case the set B∖Ann(P) is infinite; that is, there are infinitely many vertices in ΓE(P) which are not annihilator prime ideals. Let ann(z)∈Ann(P) such that deg⁡([z])<∞. Then there exists an infinite set S⊆B∖Ann(P) such that no member of S is adjacent to [z] in ΓE(P). Therefore, by Lemma 10(b), we have nbd([x])⊆nbd([z]) for all [x]∈S. Hence, using Proposition 4, it follows that the neighborhood nbd([z]) is an infinite set. This contradiction completes the proof.Given a posetP, if there is a vertex [z]∈V(ΓE(P)) such that deg⁡([z])>deg⁡([x]) for all [x]∈V(ΓE(P))∖{[z]}, then it follows from Lemma 10(a) that [z] is a maximal element of A, that is, an annihilator prime ideal of P (compare with [12, Proposition 3.1]). In this context we have a more general result given as follows (compare with [12, Proposition 3.6]).Proposition 15.
LetP be a poset such that V(ΓE(P))<∞. Then every vertex of maximal degree in ΓE(P) is a maximal element of A, that is, an annihilator prime ideal of P.Proof.
If a vertex of maximal degree inΓE(P) is not a maximal element of A, then, using Lemma 10(a) and the condition that V(ΓE(P))<∞, we have a contradiction to the maximality of its degree. Hence the result follows.Note that the converse of the above proposition is false; that is, an annihilator prime ideal need not always be of maximal degree. For example, consider the poset{∅,{1},{2},{3},{4},{5},{6},{1,2},{3,4},{3,5},{3,6}} under set inclusion, in which ann({3}) is a prime ideal but ann({1,2}) is not; however, in the corresponding reduced zero-divisor graph, the degree of ann({3}) is 6 and that of ann({1,2}) is 7.As an immediate consequence of Proposition15, we have the following result.Corollary 16.
LetP be a poset such that V(ΓE(P))<∞. Then ΓE(P) is a regular graph if and only if it is a complete graph.Proof.
In a regular graph, every vertex is of maximal degree. Therefore, ifΓE(P) is a regular graph, then, by Proposition 15, V(ΓE(P)) coincides with Ann(P). Thus, by Proposition 11(a), ΓE(P) is a complete graph. The converse is trivial.In the same context, it may be worthwhile to mention the following result.Proposition 17.
LetP be a poset. If Γ(P) is an r-regular graph, then ΓE(P) is a complete graph.Proof.
In view of Proposition11(a), it is sufficient to show that Ann(P)=V(ΓE(P)). On the contrary, suppose that there exists a vertex ann(x)∈V(ΓE(P))∖Ann(P). Since the degree of each vertex in Γ(P) is finite, P is a poset with ACC for annihilators. Therefore, there exists ann(z)∈Ann(P) such that ann(x)⊊ann(z), whence deg⁡(x)+1≤deg⁡(z). This contradicts the fact that Γ(P) is an r-regular graph, and the proposition is proved.Proposition 18.
LetP be a poset with ACC for annihilators. If |Ann(P)|<∞, then V(ΓE(P))<∞; in fact, V(ΓE(P))≤2|Ann(P)|-2.Proof.
Consider the setS of all subsets of Ann(P). Then S=2|Ann(P)|<∞. Given ann(x)∈B=V(ΓE(P)), let Sx denote the largest subset of Ann(P) such that ann(x)⊆ann(z) for all ann(z)∈Sx; in other words, Sx={ann(z)∈Ann(P)∣ann(x)⊆ann(z)}. Note that each member of B is contained in some member of Ann(P) but no member of B is contained in every member of Ann(P). Therefore, we have a well-defined map ψ:V(ΓE(P))→S given by ann(x)↦Sx such that ∅,Ann(P)∉Im⁡(ψ). It is now enough to show that ψ is a one-one map. So, let ann(x),ann(y)∈B such that Sx=Sy. Let w∈ann(x)∖{0}. Then, by Lemma 10(c) and the definition of Sx, we have ann(w)⊈ann(z) for each ann(z)∈Sx. It follows that ann(w)∪ann(y)⊈ann(z) for each ann(z)∈Sx=Sy. Also, by the maximality of Sy, we have ann(w)∪ann(y)⊈ann(z) for each ann(z)∈Ann(P)∖Sy. Therefore, using Lemma 10(c) once again, we have w∈ann(y). Thus, ann(x)⊆ann(y). Similarly, we have ann(y)⊆ann(x), and hence, [x]=[y]. This completes the proof.It may noted here that the bound mentioned in the above proposition is the best possible. For example, consider a finite setX={1,2,…,n} and define P to be the set of all subsets of X partially ordered under set inclusion with least element ϕ. It is then easy to see that the vertex set of Γ(P) consists precisely of nontrivial proper subsets of X, Ann(P)={{1},{2},…,{n}}, and ΓE(P)≅Γ(P).As an immediate consequence of Proposition18, we have the following result.Corollary 19.
LetP be a poset. Then V(ΓE(P))<∞ if and only if ω(P)<∞.Proof.
Ifω(P)<∞, then, by [6, Lemma 2.4], P is a poset with ACC for annihilators, and so it follows from Propositions 12 and 18 that V(ΓE(P))<∞. The converse is trivial, since ω(P)=ω(ΓE(P)).Note that ifP is a poset such that (x]∩Min⁡(P×)≠∅ for all x∈Z(P)×, then P is not necessarily a poset with ACC for annihilators. For example, one may look at the same poset that has been considered in the para preceding Proposition 9. However, we have the following small result concerning such posets.Proposition 20.
LetP be a poset such that (x]∩Min⁡(P×)≠∅ for all x∈Z(P)×. Then Ann(P)={ann(z)∣z∈Min⁡(P×)}.Proof.
Letz∈Min⁡(P×). Suppose that ann(z)⊆ann(x), where ann(x)∈A. Clearly, x∉ann(z), which means that z≤x. But then ann(x)⊆ann(z), and so it follows that ann(z)∈Max⁡(A)=Ann(P). Conversely, suppose that ann(x)∈Ann(P). Choose z∈(x]∩Min⁡(P×). Then, ann(x)⊆ann(z), and so it follows from the maximality of ann(x) in A that ann(x)=ann(z). This completes the proof.We conclude our discussion with the following example.Example 21.
Consider a partially ordered set(6)P=0∪A∪⋃k∈NBk,
where A is any set (finite or infinite), Bk={(1/n1,1/n2,…,1/nk)∣ni∈Nfor1≤i≤k}, and the partial order is defined as follows:
(7)a,1n1,1n2,…,1nk>0,1n1,1n2,…,1nk>1n1,1n2,…,1nk,1nk+1,…,1nk+t,1n1,1n2,…,1nk-1,1nk>1n1,1n2,…,1nk-1,1nk+1
for all a∈A and for all k,t,ni∈N with 1≤i≤k. It is not difficult to see that Ann(P)={ann(a)∣a∈A}, the set {(1/n,1/n)∣n∈N} is an infinite clique, and ann((1))⊊ann((1/2))⊊ann((1/3))⊊⋯ is an infinite strictly ascending chain of annihilator ideals in P. Thus, a poset may have finitely many annihilator prime ideals but fail to be a poset with ACC for annihilators. In this example we also have A=Min⁡(P×) but (x]∩Min⁡(P×)=∅ for all x∈Z(P)×∖Min⁡(P×), showing that the converse of Proposition 20 is far from being true.



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