L-algebras and their ideals: from simplicity to semidirect products

TL;DR

This paper characterizes ideals in semidirect products of L-algebras, introduces a family of finite simple L-algebras, and explores their structure and applications to linear Hilbert algebras.

Contribution

It provides a detailed characterization of ideals in semidirect products and introduces a new class of finite simple L-algebras with structural insights.

Findings

Characterization of ideals in semidirect products of L-algebras

Identification of a family of finite simple L-algebras

Application to linear Hilbert algebras and symmetric semidirect products

Abstract

In this paper, we investigate the ideals of semidirect products of L-algebras and the structure of simple L-algebras. We provide a precise characterization of the ideals of semidirect products and describe the structure of their prime spectrum. Furthermore, we introduce a family of finite simple L-algebras and prove that every simple linear L-algebra belongs to this family. We also show that the family we construct coincides with the class of simple algebras in a certain subclass of finite CKL-algebras. As an application, we use these results to give a clear description of linear Hilbert algebras and their symmetric semidirect products.