Categorical equivalences for root ' quasi-MV Algebras

In previous investigations into the subject {[}Giuntini et al. (2007, Studia Logica, 87, 99-128), Paoli et al. (2008, Reports on Mathematical Logic, 44, 53-85), Bou et al. (2008, Soft Computing, 12, 341-352)], square root quasi-MV algebras have been mainly viewed as preordered structures w.r.t. the induced preorder relation of their quasi-MV term reducts. In this article, we shall focus on a different relation which partially orders cartesian square root ` quasi-MV algebras. We shall prove that: (i) every cartesian square root quasi-MV algebra is embeddable into an interval in a particular Abelian l-group with operators; (ii) the category of cartesian square root quasi-MV algebras isomorphic with the pair algebras over their own polynomial MV subreducts is equivalent both to the category of such l-groups (with strong order unit), and to the category of MV algebras. As a by-product of these results we obtain a purely group-theoretical equivalence, namely between the mentioned category of l-groups with operators and the category of Abelian l-groups (both with strong order unit).