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 consid...
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| Veröffentlicht in: | The bulletin of symbolic logic 2016-06, Vol.22 (2), p.170-214 |
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| container_title | The bulletin of symbolic logic |
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| creator | CARAMELLO, OLIVIA RUSSO, ANNA CARLA |
| description | 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-interpretability 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. |
| doi_str_mv | 10.1017/bsl.2015.47 |
| format | Article |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; Cambridge University Press Journals Complete |
| subjects | Algebra Boolean Boolean data Cartesianism Classification Equivalence Equivalence relation Functors Group theory Homomorphisms Mathematical theorems Mathematical topoi Monoids Semantics Sentences Sequents |
| title | LATTICE-ORDERED ABELIAN GROUPS AND PERFECT MV-ALGEBRAS: A TOPOS-THEORETIC PERSPECTIVE |
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