On homomorphism-homogeneous point-line geometries
cytuj
pobierz pliki
RIS BIB ENDNOTEChoose format
RIS BIB ENDNOTEOn homomorphism-homogeneous point-line geometries
Publication date: 08.10.2019
Reports on Mathematical Logic, 2019, Number 54, pp. 101 - 119
https://doi.org/10.4467/20842589RM.19.007.10655Authors
On homomorphism-homogeneous point-line geometries
A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Mašulović.
Received 15 November 2018
[1] R. Akhtar and A.H. Lachlan, On countable homogeneous 3-hypergraphs, Arch. Math. Log. 34:5 (1995), 331–344.
[2] P.J. Cameron and J. Neˇsetˇril, Homomorphism-homogeneous relational structures, Combinatorics, Probability and Computing 15 (2006), 91–103.
[3] G.L. Cherlin, The classification of countable homogenous directed graphs and countable homogenous n-tournaments, Memoirs of the American Mathematical Society 621 (1998), American Math. Soc., Providence, RI, 1998.
[4] A. Devillers, Ultrahomogenous semilinear spaces, Proc. London Math. Soc. 84:3 (2002), 35–58.
[5] A. Devillers and J. Doyen, Homogenous and Ultrahomogenous Linear Spaces, Journal of Combinatorial Theory Series A 84:2 (1998), 236–241.
[6] I. Dolinka and ´ E. Jungabel, Finite homomorphism-homogeneous permutations via edge colourings of chains, The Electronic Journal of Combinatorics 19:4 (2012), # P17, (15pp.)
[7] I. Dolinka and D. Mašulović, Remarks on homomorphism-homogeneous lattices and semilattices, Monatshefte f¨ur Mathematik 164:1 (2011), 23–37.
[8] R. Fra¨ıss´e, Sur certains relations qui g´en´eralisent l'ordre des nombres rationnels, C. R. Acad. Sci. Paris 237 (1953), 540–542.
[9] A.D. Gardiner, Homogenous graphs, Journal of Combinatorial Theory (B) 20 (1976), 94–102.
[10] D. Hartman, J. Hubiˇcka, and D. Mašulović, Homomorphism-homogeneous Lcolored graphs, European Journal of Combinatorics 35 (2014), 313–323.
[11] D. Hartman and D. Mašulović, Towards finite homomorphism-homogeneous relational structures, Electronic Notes in Discrete Mathematics 38 (2011), 443–448.
[12] A. Ili´c, D. Mašulović, and U. Rajkovi´c, Finite homomorphism-homogenous tournaments with loops, Journal of Graph Theory 59:1 (2008), 45–58.
[13] ´ E. Jungabel and D. Mašulović, Homomorphism-homogeneous monounary algebras, Mathematica Slovaca 63:5 (2013), 993–1000.
[14] A.H. Lachlan, Countable homogenous tournaments, Transactions of the American Mathematical Society 284:2 (1984), 431–461.
[15] A.H. Lachlan and R.E. Woodrow. Countable ultrahomogenous undirected graphs. Transaction of the American Mathematical Society 262 (1980), 51–94.
[16] D. Mašulović, Homomorphism-homogenous partially ordered sets, Order 24:4 (2007), 215–226.
[17] D. Mašulović, Some classes of finite homomorphism-homogeneous point-line geometries, Combinatorica 33:5 (2013), 573–590.
[18] M. Rusinov and P. Schweitzer, Homomorphism-homogeneous graphs, Journal of Graph Theory 65:3 (2010), 253–262.
[19] J.H. Schmerl, Countable homogenous partially ordered sets, Algebra Universalis 9 (1979), 317–321.
Information: Reports on Mathematical Logic, 2019, Number 54, pp. 101 - 119
Article type: Original article
Titles:
On homomorphism-homogeneous point-line geometries
On homomorphism-homogeneous point-line geometries
Mathematical Institute, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary
Published at: 08.10.2019
Article status: Open
Licence: CC BY-NC-ND
Percentage share of authors:
Article corrections:
-Publication languages:
EnglishView count: 1438
Number of downloads: 1069