Is\'{e}ki spaces of semirings

TL;DR

This paper explores the topological properties of Iséki spaces associated with ideals in semirings, establishing conditions for quasi-compactness, irreducibility, and connectedness, and linking strongly connected spaces to idempotent elements.

Contribution

It provides new characterizations of Iséki spaces in semirings, including conditions for quasi-compactness and connectedness, and relates topological properties to algebraic features of semirings.

Findings

Iséki spaces are quasi-compact for Noetherian semirings.

Characterization of Iséki spaces with unique generic points.

Strongly connected Iséki spaces imply existence of non-trivial idempotents.

Abstract

The aim of this paper is to study Iseki spaces of distinguished classes of ideals of a semiring endowed with a topology. We show that every Is\'{e}ki space is quasi-compact whenever the semiring is Noetherian. We characterize Is\'{e}ki spaces for which every non-empty irreducible closed subset has a unique generic point. Furthermore, we provide a sufficient condition for the connectedness of Is\'{e}ki spaces and show that the strongly connectedness of an Is\'{e}ki space implies the existence of non-trivial idempotent elements of semirings.