Geodesic
Direct product decompositions of bounded commutative residuated $\ell$-monoids
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1129-1143 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\c A$.
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 $\c A$.
Classification : 03G25, 06D35, 06F05
Keywords: bounded commutative residuated $\ell$-monoid; lattice; direct product decomposition; internal direct factor 
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Jakubík, Ján. Direct product decompositions of bounded commutative residuated $\ell$-monoids. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1129-1143. http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a18/

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