On von Neumann regular elements in f-rings

Abstract

Let B be an Archimedean reduced f-ring. A positive element ${\omega}$ in B is said to satisfy the property ${(\ast)}$ if for every f-ring A with identity e and every ${\ell}$-group homomorphism ${\gamma : A \rightarrow B}$ with ${\gamma(e) = \omega}$, there exists a unique ${\ell}$-ring homomorphism ${\rho: B \rightarrow B}$ such that ${\gamma = \omega \rho}$ and ${\rho(e)^{\perp \perp} = \omega^{\perp \perp}}$. Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property ${(\ast)}$ and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way.