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.

The study of the interrelationship between algebra and graph theory by associating a graph to an algebraic object was initiated, in 1988, by Beck [

In 2009, Halaš and Jukl [

In this paper, inspired by the ideas of Mulay [

In this section, we put together some well-known concepts, most of which can be found in [

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

A subset of the vertex set of a graph

The

A graph

Next we turn to partially ordered sets and their zero-divisor graphs. A nonempty set is said to be a

Let

Let

Let

Finally, we would like to mention that all posets considered in this paper are with least element

In this section, we study some of the basic properties of the reduced graph of the zero-divisor graph of a poset.

Let

The graph

Given a poset

In view of Fact

By [

Let

Let

For proving the given assertions, we first note that there exist two distinct vertices

From [

Let

Assume that

The consequential statement is clear as, by the choice of

As an immediate consequence, we have the following corollary which may be compared with [

There is no poset

Our next result is a small observation, which has some interesting consequences in contrast to some results of a similar nature in [

Let

Let

The above proposition says, in other words, that the reduced zero-divisor graph of a poset cannot be further reduced (symbolically,

Let

Let

Let

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.

There is only one graph with exactly three vertices that can be realized as the graph

Since, given a poset

One of the obvious consequences of Fact

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

If

If

Our first result concerning the set of all annihilator prime ideals of a poset is given as follows.

Let

By [

Given a poset

The following lemma plays a crucial role in this section.

Let

Given

Given

Given

If

If

(a) In view of Proposition

(b) If

(c) If

(d) Let

(e) If

Let

Given

If

In view of Fact

If

Let

By Proposition

It follows immediately from Proposition

Let

Given a poset

Let

Suppose that

Given a poset

Let

If a vertex of maximal degree in

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

As an immediate consequence of Proposition

Let

In a regular graph, every vertex is of maximal degree. Therefore, if

In the same context, it may be worthwhile to mention the following result.

Let

In view of Proposition

Let

Consider the set

It may noted here that the bound mentioned in the above proposition is the best possible. For example, consider a finite set

As an immediate consequence of Proposition

Let

If

Note that if

Let

Let

We conclude our discussion with the following example.

Consider a partially ordered set

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first author wishes to express his thanks to M. R. Pournaki of Sharif University of Technology and M. Alizadeh of University of Tehran, both from Iran, for their encouragement and useful suggestions. The second author is grateful to Council of Scientific and Industrial Research (India) for its financial assistance (File no. 09/347(0209)/2012-EMR-I).