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

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