The Zero-divisor Graphs of Posets and an Application to Semigroups

On the structure of compact graphs

“…As it was defined by Lu and Wu in [7], a graph G is called a compact graph if G contains no isolated vertices and for each pair x, y of non-adjacent vertices of G, there is a vertex z with N (x) ∪ N (y) ⊆ N (z). It was proved in [7,Theorem 3.1] that a simple graph G is the zero-divisor graph of a poset if and only if G is a compact graph.…”

“…It was proved in [7,Theorem 3.1] that a simple graph G is the zero-divisor graph of a poset if and only if G is a compact graph. Therefore, compact graphs play an important role in the study of zero-divisor graphs of posets.…”

“…Therefore, compact graphs play an important role in the study of zero-divisor graphs of posets. To find some kinds of zero-divisor graphs we refer the reader to [1,[5][6][7] and [8]. In this article, we continue the study of graph-theory properties of compact graphs by determining when an arbitrary graph is a compact graph.…”

“…The following theorem from [7] reflects some properties of compact graphs and will be used in this paper, frequently. …”

The zero divisor graphs of Boolean posets

“…This concept is well studied in algebraic structures as well as in ordered structures such as rings, semigroups, lattices, semilattices, posets and qosets; see Anderson et.al. [1], LaGrange [15,16], Lu and Wu [18], Joshi and Khiste [12], Nimbhorkar et.al. [21], Halaš and Jukl [6], Joshi [11], Joshi, Waphare and Pourali [13,14] and Halaš and Länger [9].…”