Lattice-ordered matrix algebras over real GCD-domains

TL;DR

This paper proves Weinberg's conjecture for matrix algebras over GCD-domains and classifies all lattice orders on 2x2 matrix algebras over such domains, advancing understanding of algebraic order structures.

Contribution

It confirms Weinberg's conjecture for matrix algebras over GCD-domains and classifies all lattice orders on 2x2 matrix algebras over these domains.

Findings

Weinberg's conjecture is proved for all n ≥ 2.

All lattice orders on 2x2 matrix algebras over GCD-domains are classified.

The structure of lattice-ordered matrix algebras over GCD-domains is clarified.

Abstract

Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 2) $ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $ 2 \times 2 $ matrix algebra over $ R $ are obtained.