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The prime spectrum of an $L$-algebra

W. Rump, L. Vendramin

TL;DR

The paper addresses distributivity of the lattice of ideals in an arbitrary L-algebra and develops a corresponding spectral theory. It proves that the ideal lattice I(X) is distributive for every L-algebra X, enabling the application of spectrum theory to Spec(X) and linking primes to quotients that are subdirectly irreducible. It constructs the spectrum Spec(X) as the space of prime ideals of I(X), shows it is sober with a basis of quasi-compact open sets, and characterizes prime ideals via subdirect irreducibility. It analyzes how spectra behave under products and subalgebras/quotients, proving open and closed subsets of Spec(X) arise as spectra under favorable conditions and discussing quasi-prime versus prime elements, illustrating limits of spectral behavior in general.

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.

The prime spectrum of an $L$-algebra

TL;DR

The paper addresses distributivity of the lattice of ideals in an arbitrary L-algebra and develops a corresponding spectral theory. It proves that the ideal lattice I(X) is distributive for every L-algebra X, enabling the application of spectrum theory to Spec(X) and linking primes to quotients that are subdirectly irreducible. It constructs the spectrum Spec(X) as the space of prime ideals of I(X), shows it is sober with a basis of quasi-compact open sets, and characterizes prime ideals via subdirect irreducibility. It analyzes how spectra behave under products and subalgebras/quotients, proving open and closed subsets of Spec(X) arise as spectra under favorable conditions and discussing quasi-prime versus prime elements, illustrating limits of spectral behavior in general.

Abstract

We prove that the lattice of ideals of an arbitrary -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 -algebras and characterize prime ideals in topological terms.
Paper Structure (4 sections, 15 theorems, 32 equations)

This paper contains 4 sections, 15 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The ideals of an $L$-algebra
  3. The spectrum of an $L$-algebra
  4. The spectrum of special $L$-algebras

Key Result

Theorem 1

Let $X$ be an $L$-subalgebra of a self-similar $L$-algebra $A$. Then $A\cong S(X)$ if and only if the monoid $A$ is generated by $X$.

Theorems & Definitions (40)

  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Definition 1
  • Theorem 2
  • Corollary 3
  • proof
  • Proposition 1
  • proof
  • ...and 30 more