Some remarks on type n lattice-ordered algebras and a question of Huijsmans

In this paper, type n lattice-ordered algebras are introduced and a characterization is given for those of type 0 and type 1. Moreover we investigate the question: Let A be a lattice-ordered algebra with unit element \(e >0\) $e>0$ in which every positive element has an inverse. Under what conditions A is lattice and algebra isomorphic to \({\mathbb {R}}\) $\mathbb{R}$ ? We have shown that for certain algebras the question has a positive answer, generalizing thus a result of Scheffold. We also obtained a result similar to Edwards’ Theorem for normed lattice-ordered algebras.