The prime spectrum of an $L$-algebra

TL;DR

This paper establishes that the ideal lattice of any $L$-algebra is distributive, enabling a broad spectral theory and topological characterization of prime ideals, thus advancing the algebraic and topological understanding of $L$-algebras.

Contribution

It proves the distributivity of the ideal lattice of arbitrary $L$-algebras and characterizes their prime ideals topologically, extending spectral theory without restrictions.

Findings

Ideal lattice of $L$-algebras is distributive

Spectral theory applies broadly to $L$-algebras

Prime ideals are characterized topologically

Abstract

We prove that the lattice of ideals of an arbitrary $L$-algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of $L$-algebras and characterize prime ideals in topological terms.