Generalizations of pseudo MV-algebras and generalized pseudo effect algebras

Jan Kühr

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 2, page 395-415
  • ISSN: 0011-4642

Abstract

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We deal with unbounded dually residuated lattices that generalize pseudo M V -algebras in such a way that every principal order-ideal is a pseudo M V -algebra. We describe the connections of these generalized pseudo M V -algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo M V -algebra A by means of the positive cone of a suitable ℓ -group G A . We prove that the lattice of all (normal) ideals of A and the lattice of all (normal) convex ℓ -subgroups of G A are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo M V -algebra is commutative.

How to cite

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Kühr, Jan. "Generalizations of pseudo MV-algebras and generalized pseudo effect algebras." Czechoslovak Mathematical Journal 58.2 (2008): 395-415. <http://eudml.org/doc/31217>.

@article{Kühr2008,
abstract = {We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.},
author = {Kühr, Jan},
journal = {Czechoslovak Mathematical Journal},
keywords = {pseudo $MV$-algebra; $DR\ell $-monoid; generalized pseudo effect algebra; pseudo MV-algebra; DR-monoid; generalized pseudo effect algebra},
language = {eng},
number = {2},
pages = {395-415},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalizations of pseudo MV-algebras and generalized pseudo effect algebras},
url = {http://eudml.org/doc/31217},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Kühr, Jan
TI - Generalizations of pseudo MV-algebras and generalized pseudo effect algebras
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 395
EP - 415
AB - We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.
LA - eng
KW - pseudo $MV$-algebra; $DR\ell $-monoid; generalized pseudo effect algebra; pseudo MV-algebra; DR-monoid; generalized pseudo effect algebra
UR - http://eudml.org/doc/31217
ER -

References

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