## 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 ≅ ℤ^{n}, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕⋯. A few such conditions are found and a few conjectured. In studying ℝ^{n}, we encounter perpendicularity in a vector space.

Original language | English |
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Pages (from-to) | 235-247 |

Number of pages | 13 |

Journal | ACTA UNIVERSITATIS SAPIENTIAE: MATHEMATICA |

Volume | 9 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

Publication type | A1 Journal article-refereed |

## Keywords

- Abelian group
- Perpendicularity
- perpendicularity

## Publication forum classification

- Publication forum level 1