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Keywords:
bounded commutative residuated $\ell$-monoid; lattice; direct product decomposition; internal direct factor
Summary:
The notion of bounded commutative residuated $\ell$-monoid ($BCR$ $\ell$-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $\c A$ be a $BCR$ $\ell$-monoid; we denote by $\ell (\c A)$ the underlying lattice of $\c A$. In the present paper we show that each direct product decomposition of $\ell (\c A)$ determines a direct product decomposition of $\c A$. This yields that any two direct product decompositions of $\c A$ have isomorphic refinements. We consider also the relations between direct product decompositions of $\c A$ and states on $\c A$.
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