On the Complement of the Zero-Divisor Graph of a Partially Ordered Set

Abstract

In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

Intersection graphs associated with semigroup acts

“…As we mentioned in Introduction, it follows from [9,Corollary 4.3] that the intersection graph of an S-act with finite clique number is weakly perfect. In this section, we find the clique number (equivalently, chromatic number) of such graphs for some classes of S-acts.…”

“…A class of weakly perfect intersection graphs of ideals of a finite ring can be found in [15]. This result was generalized in [9], where Corollary 4.3 shows the intersection graph of submodules of any finite R-module (where R is any ring) is weakly perfect. In fact, the intersection graph of an intersection-closed family of non-empty subsets of a set is weakly perfect if it has finite clique number.…”

On Weakly Perfect Annihilating-Ideal Graphs

“…On the lines of Beck's theorem for reduced rings, Lu and Wu [18], Halaš and Jukl [12] and Joshi [13] essentially proved the following theorem for posets. Recently, Devhare et al [10] proved that the complement of the zero-divisor graph Γ c (P) of a poset P is weakly perfect when ω(Γ c (P)) < ∞. This gives us a large family of examples of weakly perfect graphs.…”

Correction to ‘On the Complement of the Zero-Divisor Graph of a Partially Ordered Set’

“…

The purpose of this note is to communicate an error in [2] and correct the results that are affected. The error lies in the proof of [2, Lemma 3.8], where it was not taken into account that edges can appear between vertices of G c (A) when passing to G c (A ∪ {a}).

…”

“…In this note, we correct these results by introducing a condition on the set A of atoms of P (although, Lemma 3.8 is not addressed since it can be omitted under the new assumptions). By using the correction to [2, Theorem 1.1], we confirm that [2, Corollaries 4.1 and 4.2] do not require revision.The following definitions will be used and all other notation is the same as in [2]. Let P be a partially ordered set with zero and Z(P) {0}.…”