665 research outputs found
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
On the geometric theory of local MV-algebras
We investigate the geometric theory of local MV-algebras and its quotients
axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We
show that, whilst the theory of local MV-algebras is not of presheaf type, each
of these quotients is a theory of presheaf type which is Morita-equivalent to
an expansion of the theory of lattice-ordered abelian groups. Di
Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the
quotient axiomatizing the local MV-algebras in Chang's variety, that is, the
perfect MV-algebras. We establish along the way a number of results of
independent interest, including a constructive treatment of the radical for
MV-algebras in a fixed proper variety of MV-algebras and a representation
theorem for the finitely presentable algebras in such a variety as finite
products of local MV-algebras.Comment: 52 page
Pseudo MV-algebras and Lexicographic Product
We study algebraic conditions when a pseudo MV-algebra is an interval in the
lexicographic product of an Abelian unital -group and an -group
that is not necessary Abelian. We introduce -perfect pseudo MV-algebras
and strong -perfect pseudo MV-algebras, the latter ones will have a
representation by a lexicographic product. Fixing a unital -group
, the category of strong -perfect pseudo MV-algebras is
categorically equivalent to the category of -groups.Comment: arXiv admin note: text overlap with arXiv:1304.074
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