Adjacency preserving mappings between point-line geometries
The author studies the adjacency preserving mapping of a certain class of point-line geometries \(G=(P,L)\) with finite diameter, where \(P\) denotes the set of points and \(L\) the set of lines [see \textit{F. Buekenhout}, Handbook of incidence geometry, North-Holland 1420 (1995; Zbl 0821.00012)]. Lines are subsets of \(P\). Two distinct points are adjacent iff they are on the line. The distance between two distinct points \(x\) and \(y\) is defined as the smallest integer \(d\) with the property that there is a sequence of \(d+1\) consecutively adjacent points \(x=x_0\), \(x_1,\dots, x_d=y\), and \(d(x,x)=0\) for all \(x\in P\). There are demanded the following properties in this article. 1. For any points \(x,y\in P\), there is a point \(z\) with \(d(x,z)=d(x,y)+d(y,z)=n\), where \(n\geq 2\) is the diameter of \(G\). 2. Every line contains at least three points. 3. For any line \(l\in L\) and any point \(x\in P\), either all points of \(l\) have the same distance from \(x\), or \(l\) contains a unique point nearest to \(x\). 4. For any \(d=1,\dots,n\) and for all points \(x,y,z\in P\) with \(d(x,y)=d\), \(d(x,z)=d-1\), \(d(y,z)=1\), there is a point \(w\) with \(d(x,w) =1\), \(d(y,w)=d-1\), \(d(z,w)=d\). 5. For any two adjacent points \(x,y\) there is a unique line \(l(x,y)\) containing \(x\) and \(y\) and any point \(z\) witch is adjacent to \(x\) and \(y\) is an element of \(l(x,y)\). 6. For any points \(x,y,z\in P\) with \(d(x,z)=n=d(y,z)\) there exist consecutively adjacent points \(x=x_0,\dots,x_k=y\) with \(d(x_i,z)=n\), and on every line \(l(x_i,x_{i+1})\) there exists a point \(p_i\) with \(d(p_i,z)=n-1\), for all \(i=0,\dots,k-1\). The main result of the article is the following theorem. Let \(G=(P,L)\), \(G'=(P',L')\) be two point-line geometries with finite diameters, which have the above properties 1--6. Let \(\varphi: P\to P'\) be a mapping which satisfies: 1. for all \(x,y\in P\), \(d(x,y)= 1\) implies \(d (x^\varphi,y^\varphi)=1\), 2. there exist \(p,q\in P\) with \(d(p,q)=d (p^\varphi,q^\varphi)=\text{diam}(G)\). Then \(\varphi\) is an isometry: \[ d(x,y)=d(x^\varphi,y^\varphi)\quad\text{for all }x,y\in P. \] Hence the geometries \((P,L)\) and \((\varphi(P),\varphi(L))\) are isomorphic, where \(\varphi(L)=\{ l^\varphi\mid l\in L\}\) and \(l^\varphi:=\{x^\varphi\mid x\in l\}\). The applications of this theorem in geometry of Hermitian matrices and in dense near polygons is given too.
scientific article; zbMATH DE number 5076686