Geodesic
Orthogonalities in the multiplicative group $\mathbb{Q}_+$
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2015), pp. 18-26 Cet article a éte moissonné depuis la source Math-Net.Ru

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The relation of orthogonality in various algebraic structures arouses the interest of mathematicians. For example, Davis proposed an interesting approach to the introduction of orthogonality in Abelian groups (Orthogonality relation on Abelian groups. Journal of the Australian Mathematical Society. Series A, vol. 19, 1975); F. Eugeni, B. Rizzi (An incidence algebra on rational numbers. Rendiconti di Matematica, vol. 12, 1979) and G. Birkhoff (Lattice theory. Providence. Rhode Island, 1965) explored orthogonality in ortholattices; Kopytov V.M. (Lattice-Ordered Groups, Nauka, Moscow, 1984) notes that the concept of orthogonality plays an important role in the whole theory of l-groups. Haukkanen and others (Perpendicularity in an Abelian group. International Journal of Mathematics and Mathematical Sciences, vol. 13, 2013) introduced the concept of orthogonality in an Abelian group with the help of axioms. The purpose of the present paper is to get some results about orthogonalities in the multiplicative Abelian group of positive rational numbers. We describe the known orthogonalities of and show that one of the relations of that was introduced in the article of Haukkanen is not an orthogonality. We construct an infinite set of new orthogonalities in by two different ways.
Keywords: orthogonality, Abelian group, divisibility in $\mathbb{Q}_+$, free Abelian group. 
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S. Ya. Grinshpon; S. L. Fukson. Orthogonalities in the multiplicative group $\mathbb{Q}_+$. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2015), pp. 18-26. http://geodesic.mathdoc.fr/item/VTGU_2015_6_a1/

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[2] Davis G., “Orthogonality relation on abelian groups”, Journal of the Australian Mathematical Society. Series A, 19 (1975), 173–179 | DOI | MR | Zbl

[3] Eugeni F., Rizzi B., “An incidence algebra on rational numbers”, Rendiconti di Matematica, 12 (1979) | MR | Zbl

[4] Birkhoff G., Lattice theory, Providence, Rhode Island, 1965

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[7] Haukkanen P., Mattila M., Merikoski J. K., Tossavainen T., “Perpendicularity in an Abelian group”, International Journal of Mathematics and Mathematical Sciences, 13 (2013) | MR

[8] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir Publ., M., 1974, 335 pp. (in Russian)