Linear Geometry: completions and projectivizations

T. Banakh; I. Hetman; A. Ravsky; V. Pshyk

doi:10.30970/ms.65.2.191-201

T. Banakh Ivan Franko National University of Lviv (Ukraine) Jan Kochanowski University in Kielce (Poland)
I. Hetman Lviv, Ukraine https://orcid.org/0009-0009-2234-5812
A. Ravsky Pidstryhach Institute for Applied Problems of Mechanics and Matematics National Academy of Sciences of Ukraine Lviv, Ukraine
V. Pshyk Ivan Franko National University of Lviv Lviv, Ukraine https://orcid.org/0009-0000-8019-5191

DOI: https://doi.org/10.30970/ms.65.2.191-201

Анотація

Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results related to completions and free projectivizations of liners. The paper is the second survey in the series of surveys that describe the contents of the monograph ``Linear Geometry and Algebra'' [1]. The first survey [2] concentrated at properties of liners that depend on flat hulls (flats, ranks, regularity, parallelity). In this paper we survey basic notions and results related to completions and free projectivizations of liners. The material covers Chapters 7, 8, 9 of the book [1].

Біографії авторів

T. Banakh, Ivan Franko National University of Lviv (Ukraine) Jan Kochanowski University in Kielce (Poland)

Ivan Franko National University of Lviv (Ukraine)
Jan Kochanowski University in Kielce (Poland)

I. Hetman, Lviv, Ukraine

Lviv, Ukraine

A. Ravsky, Pidstryhach Institute for Applied Problems of Mechanics and Matematics National Academy of Sciences of Ukraine Lviv, Ukraine

Pidstryhach Institute
for Applied Problems of Mechanics and Matematics
National Academy of Sciences of Ukraine
Lviv, Ukraine

V. Pshyk, Ivan Franko National University of Lviv Lviv, Ukraine

Ivan Franko National University of Lviv
Lviv, Ukraine

Посилання

T. Banakh, Linear Geometry and Algebra, preprint, 2025. https://arxiv.org/abs/2506.14060 https://doi.org/10.48550/arXiv.2506.14060

T. Banakh, I. Hetman, A. Ravsky, V. Pshyk, Linear Geomery: Flats, Ranks, Regularity, Parallelity, Mat. Stud. 65 (1) (2026), 74–96. https://doi.org/10.30970/ms.65.1.74-96; https://arxiv.org/abs/2511.19455

C.J. Colbourn, J.H. Dinitz (eds.), Handbook of Combinatorial Designs, Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2007. xxii+984 pp. https://doi.org/10.1201/9781003040897

P. Dembowski, Semiaffine Ebenen, Arch. Math., 13 (1962), 120–131. https://doi.org/10.1007/BF01650055

M. Hall, Projective Planes, Trans. Amer. Math. Soc., 54 (1943), 229–277. https://doi.org/10.1090/S0002-9947-1943-0008892-4

O. Kegel, A. Schleiermacher, Amalgams and Embeddings of Projective Planes, Geometriae Dedicata, 2 (1973), 379–395. https://doi.org/10.1007/BF00181482