On extensions of orthosymmetric lattice bimorphisms

Mohamed Ali Toumi

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 4, page 425-437
  • ISSN: 0862-7959

Abstract

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In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if ( A , * ) is a commutative d -algebra and A 𝔡 its Dedekind completion, then, A 𝔡 can be equipped with a d -algebra multiplication that extends the multiplication of A . Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.

How to cite

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Toumi, Mohamed Ali. "On extensions of orthosymmetric lattice bimorphisms." Mathematica Bohemica 138.4 (2013): 425-437. <http://eudml.org/doc/260641>.

@article{Toumi2013,
abstract = {In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,\ast )$ is a commutative $d$-algebra and $A^\{\mathfrak \{d\}\}$ its Dedekind completion, then, $A^\{\mathfrak \{d\}\}$ can be equipped with a $d$-algebra multiplication that extends the multiplication of $A$. Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.},
author = {Toumi, Mohamed Ali},
journal = {Mathematica Bohemica},
keywords = {$d$-algebra; $f$-algebra; lattice homomorphism; lattice bimorphism; -algebra; -algebra; lattice homomorphism; lattice bimorphism},
language = {eng},
number = {4},
pages = {425-437},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On extensions of orthosymmetric lattice bimorphisms},
url = {http://eudml.org/doc/260641},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Toumi, Mohamed Ali
TI - On extensions of orthosymmetric lattice bimorphisms
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 4
SP - 425
EP - 437
AB - In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,\ast )$ is a commutative $d$-algebra and $A^{\mathfrak {d}}$ its Dedekind completion, then, $A^{\mathfrak {d}}$ can be equipped with a $d$-algebra multiplication that extends the multiplication of $A$. Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.
LA - eng
KW - $d$-algebra; $f$-algebra; lattice homomorphism; lattice bimorphism; -algebra; -algebra; lattice homomorphism; lattice bimorphism
UR - http://eudml.org/doc/260641
ER -

References

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