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Keywords:
$d$-algebra; $f$-algebra; lattice homomorphism; lattice bimorphism
$d$-algebra; $f$-algebra; lattice homomorphism; lattice bimorphism
Summary:
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 $A$. \endgraf 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.
References:
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators. Pure and Applied Mathematics 119 Academic Press, Orlando (1985). MR 0809372 | Zbl 0608.47039
[2] Huijsmans, S. J. Bernau,C. B.: Almost $f$-algebras and $d$-algebras. Math. Proc. Camb. Philos. Soc. 107 (1990), 287-308. DOI 10.1017/S0305004100068560 | MR 1027782 | Zbl 0707.06009
[3] Boulabiar, K., Toumi, M. A.: Lattice bimorphisms on $f$-algebras. Algebra Univers. 48 (2002), 103-116. MR 1930035 | Zbl 1059.06013
[4] Boulabiar, K.: Extensions of orthosymmetric lattice bilinear maps revisited. Proc. Am. Math. Soc. 135 (2007), 2007-2009 (electronic). DOI 10.1090/S0002-9939-07-08787-4 | MR 2299473
[5] Buskes, G., Rooij, A. van: Almost $f$-algebras: Commutativity and Cauchy Schwartz inequality. Positivity 4 (2000), 227-231. DOI 10.1023/A:1009826510957 | MR 1797125
[6] Chil, E.: The Dedekind completion of a $d$-algebra. Positivity 8 (2004), 257-267. DOI 10.1007/s11117-004-1894-1 | MR 2120121
[7] Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963). MR 0171864 | Zbl 0137.02001
[8] Grobler, J. J., Labuschagne, C. C. A.: The tensor product of Archimedean ordered vector spaces. Math. Proc. Camb. Philos. Soc. 104 (1988), 331-345. DOI 10.1017/S0305004100065506 | MR 0948918 | Zbl 0663.46006
[9] Huijsmans, C. B.: Lattice-odered algebras and $f$-algebras: a survey. Positive Operators, Riesz Spaces, and Economics Proc. Conf., Pasadena/CA (USA) 1990, Stud. Econ. Theory 2 151-169 Springer, Berlin (1991). DOI 10.1007/978-3-642-58199-1_7 | MR 1307423
[10] Lipecki, Z.: Extension of vector lattice homomorphisms revisited. Indag. Math. 47 (1985), 229-233. DOI 10.1016/1385-7258(85)90010-1 | MR 0799083 | Zbl 0589.46002
[11] Luxemburg, W. A. J., Zaanen, A. C.: Riesz Spaces. Vol. I. North-Holland Mathematical Library North-Holland Publishing Company, Amsterdam (1971). MR 0511676 | Zbl 0231.46014
[12] Pagter, B. De: $f$-Algebras and Orthomorphisms. Thesis Leiden (1981).
[13] Toumi, M. A.: Extensions of orthosymmetric lattice bimorphisms. Proc. Am. Math. Soc. 134 (2006), 1615-1621. DOI 10.1090/S0002-9939-05-08142-6 | MR 2204271 | Zbl 1100.06013
[14] Zaanen, A. C.: Riesz Spaces II. North-Holland Mathematical Library 30 North-Holland Publishing Company, Amsterdam (1983). MR 0704021 | Zbl 0519.46001
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