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Generalized MV-algebras. (English) Zbl 1063.06008
The authors define a generalized MV-algebra (GMV-algebra for short) as a residuated lattice satisfying the identities \(x/((x\vee y)\setminus x)=x\vee y=(x/((x\vee y))\setminus x\). A closure operator \(\gamma\) on a residuated lattice \(\mathbf L\) such that \(\gamma(a)\gamma(b)\leqq \gamma(ab)\) for all \(a,b\in L\) is called a nucleus on \(L\); the image \(L_\gamma\) of \(\gamma\) is endowed with a residuated lattice structure \(\mathbf L_\gamma=(L,\wedge,\vee_\gamma, \circ_\gamma, \setminus, /, \gamma(e))\), where \(\gamma(a)\vee_\gamma \gamma(b)= \gamma(a\vee b)\) and \(\gamma(a)\circ_\gamma \gamma(b)=\gamma(ab)\).
The fundamental result of the paper is the following theorem: A residuated lattice \(\mathbf M\) is a GMV-algebra if and only if there are residuated lattices \(\mathbf G, \mathbf L\), such that \(\mathbf G\) is an \(\ell\)-group, \(\mathbf L\) is the negative cone of an \(\ell\)-group, \(\gamma\) is a nucleus on \(\mathbf L\) and \(\mathbf M=\mathbf G\oplus \mathbf L_\gamma\) (where \(\oplus\) denotes the operation of the direct sum). As a consequence, the authors obtain a categorical equivalence that generalizes the results of Mundici and Dvurečenskij concerning the functor \(\Gamma\). Further, they prove that the equational theory of the variety of GMV-algebras is decidable.

MSC:
06D35 MV-algebras
06F15 Ordered groups
03B25 Decidability of theories and sets of sentences
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