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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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