A categorical equivalence for odd or even involutive FL${}_e$-chains

We exhibit a categorical equivalence between the class of odd or even involutive FL${}_e$-chains and a class of direct systems of abelian $o$-groups. Restricting this equivalence only to odd or only to even involutive FL${}_e$-chains or to further subclasses thereof (e.g., to Sugihara chains) yields further categorical equivalences.