| Popis: |
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 T1 subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by GI(P), and characterize its diameter. 1. Preliminaries Throughout this paper, (P,≤) denotes a poset with a least element 0, and all prime and maximal semi-ideals of P are assumed to be proper. For M ⊆ P, let (M) l := {x ∈ P : x ≤ m for all m ∈ M} denote the lower cone of M in P, and dually let (M) u := {x ∈ P : m ≤ x for all m ∈ M} be the upper cone of M in P. For A,B ⊆ P, we write (A,B) l instead of (A ∪ B) l and |
