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Projectable \(\ell\)-groups and algebras of logic: categorical and algebraic connections. (English) Zbl 1345.06013
Generalized MV algebras, or GMV algebras for short, are “simultaneous generalizations of MV algebras to the noncommutative, unbounded and nonintegral case”, while IGMV algebras are integral GMV algebras. In the paper [N. Galatos and C. Tsinakis, J. Algebra 283, No. 1, 254-291 (2005; Zbl 1063.06008)] it was proved in fact that “the categories of IGMV algebras and of negative cones of \(l\)-groups with a dense nucleus are equivalent”.
This paper answers the natural conjecture “that such an equivalence restricts to an equivalence of the subcategories whose objects are the projectable members of these classes of algebras”. The authors prove that, indeed, “the categories of projectable IGMV algebras and of negative cones of projectable \(l\)-groups with a dense nucleus are equivalent”.
Moreover, by adding the Gödel implication to an IGMV algebra, they introduce the notion of Gödel GMV algebra – as an algebra \((M,\wedge,\vee,\cdot,\backslash,/,\to,1)\) of type \((2,2,2,2,2,2,0)\) such that \((M,\wedge,\vee,\cdot,\backslash,/,1)\) is an IGMV algebra and \((M,\wedge,\vee,\cdot,\to,1)\) is a Gödel algebra. And they prove that there is an adjunction between the category of Gödel GMV algebras and a certain category.
Reviewer’s remarks: The readability of the paper is restricted by the use of the divisions (the “Chinese sticks”) \(\backslash\), \(/\), instead of the implications \(\to\), \(\rightsquigarrow\) (\(y\to z=z/y\) and \(y\rightsquigarrow z=y\backslash z\)) coming from logic.

MSC:
06F15 Ordered groups
06D35 MV-algebras
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G10 Logical aspects of lattices and related structures
06B20 Varieties of lattices
06F05 Ordered semigroups and monoids
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