Orthogonalities in $\mathbb{Z}_p\oplus\mathbb{Z}_p$

S. L. Fukson

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2015), pp. 46-54

Résumé

More and more geometric concepts penetrate into the space of the study of algebra. On March 2013, the International Journal of Mathematics and Mathematical Sciences carried an article titled “Perpendicularity in an Abelian Group” written by Haukkanen et al. The main objective of that article was to introduce the concept of a binary relation of orthogonality in an arbitrary Abelian group. The ratio of orthogonality in different algebraic structures has already aroused the interest of mathematicians before. For example, orthogonality in the rings was studied by Davis in the paper “Rings with orthogonality relation” (Bulletin of the Australian Mathematical Society, vol. 4, 1971), Veksler considers orthogonality in lattices and lattice-ordered groups in “Linear spaces with disjoint elements and their conversion into vector lattices” (Leningrad. Gos. Ped. Inst. Uchen. Zap., vol. 328, pp. 19–43, 1967). The purpose of the present paper is to get some results about orthogonalities for specific Abelian groups. We have investigated in detail the orthogonalities of a direct sum of cyclic groups $\mathbb{Z}_p\oplus\mathbb{Z}_p$. On the way to answer how many and which orthogonalities this group has, we consider special cases, namely, $\mathbb{Z}_3\oplus\mathbb{Z}_3$, $\mathbb{Z}_5\oplus\mathbb{Z}_5$ and only after that we are able to summarize the results to an arbitrary group $\mathbb{Z}_p\oplus\mathbb{Z}_p$. Haukkanen and others introduced the concept of orthogonality in an Abelian group with the help of axioms which are absolutely natural if we give a geometric interpretation to them. Let $G=(G,+)$ be an additive Abelian Group. Let $\perp$ be a binary relation in $G$ satisfying: $$ \begin{aligned} \text{(A1)}\quad & \forall a\in G : \exists b\in G: a\perp b,\\ \text{(A2)}\quad &\forall a\in G\setminus\{0\}: a\not\perp a,\\ \text{(A3)}\quad &\forall a, b\in G : a\perp b\rightarrow b\perp a,\\ \text{(A4)}\quad &\forall a, b, c\in G : a\perp b\land a\perp c \rightarrow a\perp (b+c),\\ \text{(A5)}\quad &\forall a, b\in G : a\perp b\rightarrow a\perp -b. \end{aligned} $$ We call $\perp$ an orthogonality in $G$. Definition. We call $\perp$ maximal orthogonality in $G$ if it satisfies the axioms (А1)–(А5) and after adding thereto any of the other pair, the ratio obtained is not orthogonality in $G$. We explore orthogonalities of groups $\mathbb{Z}_3\oplus\mathbb{Z}_3$, $\mathbb{Z}_5\oplus \mathbb{Z}_5$ and consider the general case of a direct sum $\mathbb{Z}_p\oplus\mathbb{Z}_p$. With the help of our research, we were able to find the number of elementary, maximal, and all possible orthogonalities in the studied groups. We present our results in the following table.

Détail

Keywords: Orthogonality, Abelian group, direct sum, cyclic group.
Mots-clés : group

Bibliographie

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

[2] Haukkanen P., Merikoski J. K., Tossavainen T., “Axiomatizing perpendicularity and parallelism”, Journal for Geometry and Graphics, 15:2 (2011), 129–139 | MR | Zbl

[3] Davis G., “Rings with orthogonality relation”, Bulletin of the Australian Mathematical Society, 4 (1971), 163–178 | DOI | MR | Zbl

[4] Veksler A. I., “Linear spaces with disjoint elements and their conversion into vector lattices”, Uchenye zapiski Leningradskogo gosudarstvennogo pedagogicheskogo instituta imeni A. I. Gertsena, 1967, no. 328, 19–43 (in Russian) | MR