Categorical Equivalences for '~(1/2) quasi-MV Algebras

摘要

In previous investigations into the subject [Giuntini et al. (2007, Stadia Logica, 87, 99-128), Paoli et al. (2008, Reports on Mathematical Logic, 44, 53-85), Bou et al. (2008, Soft Computing, 12, 341-352)], '~(1/2) 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 '~(1/2) quasi-MV algebras. We shall prove that: (ⅰ) every cartesian '~(1/2) quasi-MV algebra is embeddable into an interval in a particular Abelian ￡-group with operators; (ⅱ) the category of cartesian '~(1/2) quasi-MV algebras isomorphic with the pair algebras over their own polynomial MV subreducts is equivalent both to the category of such ￡-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 ￡-groups with operators and the category of Abelian ￡-groups (both with strong order unit).