Categorical equivalences for p 0 quasi - MV algebras

In previous investigations into the subject ([4], [9], [1]), p 0 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 paper, we shall focus on a di¤erent relation which partially orders cartesian p 0 quasi-MV algebras. We shall prove that: a) every cartesian p 0 quasi-MV algebra is embeddable into an interval in a particular Abelian `-group with operators; b) the category of cartesian p 0 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 byproduct 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). 1 Introduction p 0 quasi-MV algebras were introduced in [4] as the variety generated by an algebra over the complex numbers which is the abstract counterpart, in an appropriate sense, of the algebra whose universe is the set of all qumixes (i.e. density operators) of C and whose operations correspond to some of the most signi…cant quantum logical gates: see [8] for an introduction to the basic concepts of quantum computation, and [4], [6] and especially [5], for a thorough motivational introduction to our approach. Subsequent papers elaborated by our research group ([9], [1]) tried to investigate in greater detail the structure theory and the algebraic properties of this variety. Two problems, however, remained open, namely: While p 0 quasi-MV algebras bear a close resemblance to Chang’s MV algebras, as their denomination explicitly suggests, the latter thanks to Daniele Mundici’s Gamma functor ([2], Chapters 2 and 7) admit of a telling and mathematically powerful representation in terms of intervals in