Poset properties with respect to semi-ideal based zero-divisor graph

In this paper, we discuss some properties of poset P determined by semi-ideal based zero-divisor graph G_K(P), for a semi-ideal K of P. We investigate some interesting properties if G_K(P) contains a cycle. Further, we prove that if V(G_K(P)) is a generalized tree, then (V(G_K(P))\backslash S_x)\cup K and (V(G_K(P))\backslash S_{\geq x})\cup K are prime semi-ideals of P.

Keywords:

Posets , semi-ideals , prime semi-ideals , cycle , and neighborhood