Abstract

Lattice-ordered abelian groups, or abelianℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call DLMV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{DLMV}$$\end{document}. In this work we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete.