Abstract
Lattice-ordered abelian groups, or abelian \(\ell \)-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 \(\textsf{DLMV}\). 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.
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Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Giustarini and Ugolini acknowledge partial support by the MOSAIC project (H2020-MSCA-RISE-2020 Project 101007627). Ugolini acknowledges support from the Ramón y Cajal programme RyC2021-032670-I.
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Presented by Francesco Paoli; Received October 6, 2023.
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Giustarini, V., Manfucci, F. & Ugolini, S. Free Constructions in Hoops via \(\ell \)-Groups. Stud Logica (2024). https://doi.org/10.1007/s11225-024-10128-y
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DOI: https://doi.org/10.1007/s11225-024-10128-y