Ordered groups with a modality

• Autores: Franco Montagna , Constantine Tsinakis
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 211, Nº 2, 2007, págs. 511-531
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2007.01.013
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• Resumen

  • Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable modal operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between [II MTL] -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a "quantale like" category of algebras