<abstract xmlns="http://www.w3.org/1999/xhtml">We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.</abstract>

We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

Maximal perpendicularity in certain Abelian groups

Department of Art, Communication and Education

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Maximal perpendicularity in certain Abelian groups"}

We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of...

Maximal perpendicularity in certain Abelian groups

Published Online: Aug 05, 2017

Page range: 235 - 247

Received: May 24, 2016

DOI: https://doi.org/10.1515/ausm-2017-0016

Keywords
Abelian group, perpendicularity

© 2017 Mika Mattila et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Maximal perpendicularity in certain Abelian groups

Published Online: Aug 05, 2017

Page range: 235 - 247

Received: May 24, 2016

DOI: https://doi.org/10.1515/ausm-2017-0016

KeywordsAbelian group, perpendicularity

© 2017 Mika Mattila et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Abelian group, perpendicularity