On semisubtractive ideals of semirings

TL;DR

This paper investigates the structure and properties of semisubtractive ideals in semirings, introducing new subclasses, topological perspectives, and establishing their relationships with other ideal types and congruences.

Contribution

It introduces the concepts of Golan closures, $s$-strongly irreducible and $s$-irreducible ideals, and explores their properties, representations, and topological structures in semirings.

Findings

Semisubtractive ideals form a complete modular lattice.

The space of semisubtractive ideals is $T_0$, sober, connected, and quasi-compact.

A bijection exists between $s$-congruences and semisubtractive ideals.

Abstract

Our aim in this paper is to explore semisubtractive ideals of semirings. We prove that they form a complete modular lattice. We introduce Golan closures and prove some of their basic properties. We explore the relations between $Q$-ideals and semisubtractive ideals of semirings, and also study them in $s$-local semirings. We introduce two subclasses of semisubtractive ideals: $s$-strongly irreducible and $s$-irreducible, and provide various representation theorems. By endowing a topology on the set of semisubtractive ideals, we prove that the space is $T_0$, sober, connected, and quasi-compact. We also briefly study continuous maps between semisubtractive spaces. We construct $s$-congruences and prove a bijection between these congruences and semisubtractive ideals.