Communications of the
Korean Mathematical Society CKMS

The structure of a poset $P$ with smallest element $0$ is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that $Spec(P),$ the set of all prime semi-ideals of $P,$ is a compact space and $Max(P),$ the set of all maximal semi-ideals of $P,$ is a compact $T_1$ subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset $P, $ denoted by $G_I(P),$ and characterize its diameter.

Keywords: posets, semi-ideals, prime semi-ideals, zero-divisor graph

MSC numbers: 05C99, 06B35