Orthoproducts and f-representations of archimedean $${\ell}$$ ℓ -groups

Abstract

Let G be an archimedean $${\ell}$$ -group. By an f-representation of G we mean an orthomorphism-valued group homomorphism S on G for which (Sf)g = (Sg)f for all $${f, g \in G}$$ . We prove that the set $${\mathfrak{Rep}(G)}$$ of all f-representations in G is an archimedean $${\ell}$$ -group with respect to pointwise addition and ordering. Furthermore, we define an orthoproduct on G to be a bilinear map on G which is an orthomorphism in each variable separately. It turns out that the set $${\mathfrak{Opro}(G)}$$ is an archimedean $${\ell}$$ -group G with the set $${\mathfrak{Mult}(G)}$$ of f-multiplications in G as a positive cone. Moreover, we show that $${\mathfrak{Opro}(G)}$$ and $${\mathfrak{Rep}(G)}$$ are isomorphic as $${\ell}$$ -groups. In spite of that, we get a representation theorem for f-multiplications in an $${\ell}$$ -subgroup of an archimedean f-ring R with unit element. This allows us to find an example of an archimedean $${\ell}$$ -group with no nontrivial structure of an f-ring and another which cannot be a reduced f-ring.