Archimedean ℓ-algebras with multiplication closed bands

In this paper, we introduce the notion of b-algebras and we give some related properties. To be more precise, we call a b-algebra any lattice-ordered algebra A the bands of which are closed under multiplication. We obtain that A can be identified with the reals whenever A is an Archimedean b-algebra with unit element e> 0 and such that every positive element has an inverse. This improves a result by Huijsmans who got the same conclusion for f-algebras imposing the extra condition of positivity of inverses. Moreover, we show that the order continuous bidual (A∼)n∼ of an Archimedean b-algebra A is a b-algebra with respect to the Arens multiplication. Furthermore, if the b-algebra A has positive squares, then the order bidual A∼∼ is again a b-algebra.