an:06581943
Zbl 1345.06013
Gil-F??rez, Jos??; Ledda, Antonio; Paoli, Francesco; Tsinakis, Constantine
Projectable \(\ell\)-groups and algebras of logic: categorical and algebraic connections
EN
J. Pure Appl. Algebra 220, No. 10, 3514-3532 (2016).
00355455
2016
j
06F15 06D35 03B47 03G10 06B20 06F05
generalized MV algebras; integral GMV algebras; projectable IGMV algebras; negative cones of projectable \(l\)-groups; G??del GMV algebras; equivalences of categories
\textit{Generalized MV algebras}, or \textit{GMV algebras} for short, are ``simultaneous generalizations of MV algebras to the noncommutative, unbounded and nonintegral case'', while \textit{IGMV algebras} are integral GMV algebras. In the paper [\textit{N. Galatos} and \textit{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 G??del implication to an IGMV algebra, they introduce the notion of \textit{G??del 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 G??del algebra. And they prove that there is an adjunction between the category of G??del 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.
Afrodita Iorgulescu (Bucharest)
Zbl 1063.06008