PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 6, Pages 1615–1621 S 0002-9939(05)08142-6 Article electronically published on December 5, 2005 EXTENSIONS OF ORTHOSYMMETRIC LATTICE BIMORPHISMS MOHAMED ALI TOUMI (Communicated by Joseph A. Ball) Abstract. Let E be an Archimedean vector lattice, let E d be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ 0 : E × E → B is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism Ψ : E d × E d → B that not just extends Ψ 0 but also has to be orthosymmetric. As an application, we prove the following: Let A be an Archimedean d-algebra. Then the multiplication in A can be extended to a multiplication in A d , the Dedekind completion of A, in such a fashion that A d is again a d-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990. 1. Introduction The standard extension theorem under a domination hypothesis is, of course, the classical Hahn-Banach theorem. Its method of proof was used almost immediately by L.V. Kantorovich [9] to prove that a positive linear map from a dominating subspace E of a vector lattice A (i.e., for each x ∈ A there exists y ∈ E such that |x|≤ y) into a Dedekind complete vector lattice B can be extended to a positive linear of A into B. Moreover, Luxemburg and Schep [16] showed that vector lattice homomorphism of a dominating vector sublattice extends not just positively but so as to preserve the lattice operations. They did show, however, that the vector lattice homomorphic extensions were extreme points of the set of positive extensions. Lipecki, alone and with coauthors [10, 11, 12, 13, 14, 15], took up the question of describing the extreme points of the set of positive extensions. They obtained a fairly simple characterization of them, in [10], and used this to produce vector lattice homomorphic extensions. Surprisingly enough, to the best of our knowledge, no attention has been paid in the literature to the corresponding problem of the bilinear map, except in the paper of Grobler and Labuschagne [7]. Here they proved that if E and F are majorizing vector sublattices of the vector lattices A and B and if C is a Dedekind complete vector lattice, then every lattice bimorphism Ψ 0 : E × F → C can be extended to a lattice bimorphism Ψ to A × B into C. The question arises as to whether Ψ still satisfies the property (AF) when Ψ 0 has in addition the property (AF). The answer is affirmative (Theorem 1). Received by the editors February 10, 2004 and, in revised form, January 13, 2005. 2000 Mathematics Subject Classification. Primary 06F25, 47B65. Key words and phrases. d-algebra, f -algebra, lattice homomorphism, lattice bimorphism. The author thanks Professor S. J. Bernau for providing the bibliographic information of [2]. c 2005 American Mathematical Society Reverts to public domain 28 years from publication 1615 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use