A note on the commutativity of Archimedean almost \(f\)-rings (Q2746295)

A lattice-ordered ring \(R\) is called an almost \(f\)-ring if for any \(f,g\in R,\) \(f\wedge g =0\) implies \(fg = 0.\) Let \(F\) and \(G\) be groups, \(A: G\times G\rightarrow F\) is called a bimorphism if \(A\) is a group homomorphism in each coordinate. For Archimedean \(l\)-groups \(F\) and \(G\), a map \(A: G\times G\rightarrow F\) is called orthosymmetric if the following hold:NEWLINENEWLINENEWLINE(i) if \(f,g\in G\) and \(f\wedge g = 0,\) then \(A(f,g) = 0,\) and NEWLINENEWLINENEWLINE(ii) \(A\) is a group bimorphism. NEWLINENEWLINENEWLINEThe main result of this paper is to prove that every Archimedean almost \(f\)-ring is commutative. As we know, this result is not new. \textit{S. J. Bernau} and \textit{C. B. Huijsmans} [Math. Proc. Camb. Philos. Soc. 107, 287-308 (1990; Zbl 0707.06009)] have already proved that every almost \(f\)-algebra is commutative and the proof of the commutativity of almost \(f\)-algebras can be carried over to the setting of almost \(f\)-rings. However, the authors offer a different way to verify the commutativity of almost \(f\)-rings using orthosymmetric maps.