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Ordered groups with a conucleus. (English) Zbl 1185.06012
Authors’ abstract: 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 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 $$\Pi$$-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.

MSC:
 06F05 Ordered semigroups and monoids 06D35 MV-algebras 06F15 Ordered groups 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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