On reduced zero-divisor graphs of posets

In this paper we study some of the basic properties of a graph which is constructed from the equivalence classes of non-zero zero-divisors determined by 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.


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 zerodivisor graphs associated to rings and other algebraic structures (see, e.g., [2][3][4][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][8][9]. However, the zerodivisor 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.

Prerequisites
In this section, we put together some well-known concepts, most of which can be found in [13][14][15][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. Let be a graph and , ∈ ( ) the vertex set of . Then, and are said to be adjacent if ̸ = and there is an edge − between and . A walk between and is a sequence of adjacent vertices, often written as − 1 − 2 −⋅ ⋅ ⋅− − . A walk between and is called path if the vertices in it are all distinct (except, possibly, and ). A path between and is called a cycle if = . The number of edges in a path or a cycle is called its length. If ̸ = , then the minimum of the lengths of all paths between and in is called the distance between and in and is denoted by dist( , ). If there is no path between and , then we define dist( , ) = ∞. The maximum of all possible distances in is called the diameter of and is denoted by diam( ). The girth of a graph is the minimum of the lengths of all cycles in and is denoted by girth( ). If is acyclic, that is, if has no cycles, then we write girth( ) = ∞. A cycle graph is a graph that consists of a single cycle (an -gon).
A subset of the vertex set of a graph is called a clique of if it consists entirely of pairwise adjacent vertices. The least upper bound of the sizes of all the cliques of is called the clique number of and is denoted by ( ).
The neighborhood of a vertex in a graph , denoted by nbd( ), is defined to be the set of all vertices adjacent to while the degree of in , denoted by deg( ), is defined to be the number of vertices adjacent to , and so deg( ) = |nbd( )|. If deg( ) = 1, then is said to be an end vertex in . If deg( ) = deg( ) for all , ∈ ( ), then the graph is said to be a regular graph.
A graph is said to be connected if there is a path between every pair of distinct vertices in . A graph is said to be complete if there is an edge between every pair of distinct vertices in . We denote the complete graph with vertices by . An -partite graph, ≥ 2, is a graph whose vertex set can be partitioned into disjoint parts in such a way that no two adjacent vertices lie in the same part. Among the -partite graphs, the complete -partite graph is the one in which two vertices are adjacent if and only if they lie in different parts. The complete -partite graph with parts of size 1 , 2 , . . . , is denoted by 1 , 2 ,..., . The 2-partite and complete 2-partite graphs are popularly known as bipartite and complete bipartite graphs, respectively. A bipartite graph of the form 1, is also known as a star graph.
Next we turn to partially ordered sets and their zerodivisor graphs. A nonempty set is said to be a partially ordered set (in short, a poset) if it is equipped with a partial order, that is, a reflexive, antisymmetric, and transitive binary relation. It is customary to denote a partial order by "≤. " Let be a nonempty subset of a poset . If there exists ∈ such that ≤ for every ∈ , then is called the least element of . The least element of , if exists, is usually denoted by 0. An element ∈ is called a minimal element of if ∈ and ≤ imply that = . We denote the set of minimal elements of by Min( ).
Let be a poset with least element 0. An element ∈ is called a zero-divisor of if there exists ∈ × := \ {0} such that the set ( , ) := { ∈ | ≤ and ≤ } = {0}. We denote the set of zero-divisors of by ( ) and write ( ) × := ( ) \ {0}. By an ideal of we mean a nonempty subset of such that ∈ whenever ≤ for some ∈ . We say that the ideal is proper if ̸ = . For each ∈ , it is easy to see that the set ( ] := { ∈ | ≤ } is an ideal of , called the principal ideal of generated by . Given ∈ , the annihilator of in is defined to be the set ann( ) := { ∈ | ( , ) = {0}}, which is also an ideal of . Note that ∉ ann( ) for all ∈ × . A proper ideal p of is called a prime ideal of if, for every , ∈ , ( , ) ⊆ p implies that either ∈ p or ∈ p. A prime ideal p of is said to be an annihilator prime ideal (or, an associated prime) if there exists ∈ such that p = ann( ). Two annihilator prime ideals ann( ) and ann( ) are distinct if and only if ( , ) = {0} (see [6,Lemma 2.3]). We write Ann( ) to denote the set of all annihilator prime ideals of .
Finally, we would like to mention that all posets considered in this paper are with least element 0 and have nonzero zero-divisors, unless explicitly written otherwise.

Reduced Graph of Γ( ): Basic Properties
In this section, we study some of the basic properties of the reduced graph of the zero-divisor graph of a poset.
Let be a poset with least element 0 and with ( ) × ̸ = 0.
In analogy with [11,12], the graph of equivalence classes of zerodivisors of may be defined to be the graph Γ ( ) in which the vertex set is the set of all equivalence classes of the elements of ( ) × , and two vertices [ ] and [ ] are adjacent if and only if ( , ) = {0}, that is, if and only if and are adjacent in Γ( ). Note that two adjacent vertices in Γ( ) represent two distinct equivalence classes and, hence, two distinct vertices in Γ ( ). Thus Γ ( ) is also a simple graph. It may be recalled (see [12]) that, in case of a commutative ring with unity, two adjacent vertices in Γ( ) do not necessarily represent two distinct vertices in Γ ( ).
Thus, the graph Γ ( ) is the same as the reduced graph of Γ( ) considered by Lu and Wu in [10]. In particular, Γ ( ) is also a zero-divisor graph of some poset, and, hence, there are lots of structural similarities between Γ( ) and Γ ( ). There are however some more features of Γ ( ) which are worth looking into.
The graph Γ ( ) has some advantages over the zerodivisor graph Γ( ). In many cases Γ ( ) is finite when Γ( ) is infinite. Another important aspect of Γ ( ) is its connection to the annihilator prime ideals of the poset , 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. In view of Fact 1, it is easy to see that, given a poset , we have ( ) = (Γ ( )); that is, the clique numbers of Γ( ) and Γ ( ) are the same.
By [10, Corollary 3.3 (2)], we know that Γ ( ) is also a zero-divisor graph of some poset. Therefore, in view of [9, Theorem 3.3], Γ ( ) is a connected graph with diam Γ ( ) ≤ 3. Our first result of this section not only generalizes Fact 1 but also shows some similarity between Γ( ) and Γ ( ) as far as their diameter is concerned. Proof.
, and, hence, the equality holds.
For proving the given assertions, we first note that there exist two distinct vertices [ ] and [ ] of Γ ( ) such that dist([ ], [ ]) = diam(Γ ( )). Therefore, it follows from the first half of this proposition that we always have diam(Γ ( )) ≤ diam(Γ( )). However, for the reverse inequality it is not enough to have two distinct vertices , ∈ (Γ( )) such that dist( , ) = diam(Γ( )); we must also have an additional requirement; namely, ann( ) ̸ = ann( ). If dist( , ) = 3, then this additional requirement is guaranteed by the existence of a path of the form − − − in Γ( ). 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).
The consequential statement is clear as, by the choice of , we always have | (Γ ( ))| ≥ 2.
As an immediate consequence, we have the following corollary which may be compared with [12, Proposition 1.8].

Corollary 3. There is no poset for which Γ ( ) 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]. 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 graph Γ ( ) for some poset , and it is the cycle/complete graph 3 .
Proof. Since, given a poset , the graph Γ ( ) is a connected but not a star graph, we need only to note that the graph of

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.
Let be a poset. We say that is a 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( ) | ∈ × } under set inclusion. Equivalently, is a poset with ACC for annihilators if and only if every nonempty subset of A has a maximal element. Thus, if 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).
If is a poset such that ( ) < ∞, then from [6, Lemma 2.4] it follows that is a poset with ACC for annihilators. In particular, if is a poset such that deg( ) < ∞ for all ∈ (Γ( )), then is a poset with ACC for annihilators; noting that Γ( ) has no infinite clique and so, by [ which means that is a poset without ACC for annihilators.

Remark 8.
If is a poset with ACC for annihilators, then the clique number of Γ( ) need not be finite; that is, Γ( ) may contain an infinite clique; for example, consider the poset = {0}∪{{ } | ∈ 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. As such, we may treat B as the vertex set of Γ ( ). In view of this, with a slight abuse of terminology, we sometimes refer to [ ] ∈ (Γ ( )) as an annihilator ideal (resp., an annihilator prime ideal) if we have ann( ) ∈ B (resp., ann( ) ∈ Ann( )). All the forthcoming results of this section are under this identification.
(e) If ann( ) ∈ B \ Ann( ), then, by Proposition 9, there exists ann( ) ∈ A such that ann( ) ⊊ ann( ), and so we may choose V ∈ ann( ) \ ann( ) to complete the proof. Proof. In view of Fact 1, 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, If is a poset with ( ) < ∞, then, in view of [6, Lemmas 2.6], it follows from Proposition 9 that |Ann( )| < ∞. In this context, we have a stronger result in the following form. Proof. If a vertex of maximal degree in Γ ( ) is not a maximal element of A, then, using Lemma 10(a) and the condition that | (Γ ( ))| < ∞, 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. As an immediate consequence of Proposition 15, we have the following result. Proof. In a regular graph, every vertex is of maximal degree. Therefore, if Γ ( ) is a regular graph, then, by Proposition 15, (Γ ( )) coincides with Ann( ). Thus, by Proposition 11(a), Γ ( ) is a complete graph. The converse is trivial.
In the same context, it may be worthwhile to mention the following result.
Proposition 17. Let be a poset. If Γ( ) is an -regular graph, then Γ ( ) is a complete graph.