Direct product decompositions of bounded commutative residuated ℓ -monoids

Ján Jakubík

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1129-1143
  • ISSN: 0011-4642

Abstract

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The notion of bounded commutative residuated ℓ -monoid ( B C R ℓ -monoid, in short) generalizes both the notions of M V -algebra and of B L -algebra. Let A ̧ be a B C R ℓ -monoid; we denote by ℓ ( A ̧ ) the underlying lattice of A ̧ . In the present paper we show that each direct product decomposition of ℓ ( A ̧ ) determines a direct product decomposition of A ̧ . This yields that any two direct product decompositions of A ̧ have isomorphic refinements. We consider also the relations between direct product decompositions of A ̧ and states on A ̧ .

How to cite

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Jakubík, Ján. "Direct product decompositions of bounded commutative residuated $\ell $-monoids." Czechoslovak Mathematical Journal 58.4 (2008): 1129-1143. <http://eudml.org/doc/37891>.

@article{Jakubík2008,
abstract = {The notion of bounded commutative residuated $\ell $-monoid ($BCR$$\ell $-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $A̧$ be a $BCR$$\ell $-monoid; we denote by $\ell (A̧)$ the underlying lattice of $A̧$. In the present paper we show that each direct product decomposition of $\ell (A̧)$ determines a direct product decomposition of $A̧$. This yields that any two direct product decompositions of $A̧$ have isomorphic refinements. We consider also the relations between direct product decompositions of $A̧$ and states on $A̧$.},
author = {Jakubík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {bounded commutative residuated $\ell $-monoid; lattice; direct product decomposition; internal direct factor; bounded commutative residuated -monoid; lattice; direct product decomposition; internal direct factor},
language = {eng},
number = {4},
pages = {1129-1143},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Direct product decompositions of bounded commutative residuated $\ell $-monoids},
url = {http://eudml.org/doc/37891},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Jakubík, Ján
TI - Direct product decompositions of bounded commutative residuated $\ell $-monoids
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1129
EP - 1143
AB - The notion of bounded commutative residuated $\ell $-monoid ($BCR$$\ell $-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $A̧$ be a $BCR$$\ell $-monoid; we denote by $\ell (A̧)$ the underlying lattice of $A̧$. In the present paper we show that each direct product decomposition of $\ell (A̧)$ determines a direct product decomposition of $A̧$. This yields that any two direct product decompositions of $A̧$ have isomorphic refinements. We consider also the relations between direct product decompositions of $A̧$ and states on $A̧$.
LA - eng
KW - bounded commutative residuated $\ell $-monoid; lattice; direct product decomposition; internal direct factor; bounded commutative residuated -monoid; lattice; direct product decomposition; internal direct factor
UR - http://eudml.org/doc/37891
ER -

References

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  10. Jakubík, J., Csontóová, M., Cancellation rule for internal direct product decompositions of a connected partially ordered set, Math. Bohenica 125 (2000), 115-122. (2000) MR1752083
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