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Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Authors:	Helmut Karzel Jaros?aw Kosiorek Andrzej Matra?

Institution:	(1) Zentrum Mathematik, T.U. München, D-80290 München, Germany;(2) Department of Mathematics and Informatics, UWM Olsztyn, Żołnierska 14, PL-10-561 Olsztyn, Poland

Abstract:	Let E be a non empty set, let P : = E × E, ${\mathfrak{G}_{1}}$ := {x × E\|x ∈ E}, ${\mathfrak{G}_{2}}$ := {E × x\|x ∈ E}, and $\mathfrak{C}$ := {C ∈ 2^P \|∀X ∈ $\mathfrak{G}_{1} \cup \mathfrak{G}_{2}$ : \|C ∩ X\| = 1} and let $\mathfrak{B} \subseteq \mathfrak{C}$ . Then the quadruple $(P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{B})$ resp. $(P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{C})$ is called chain structure resp. maximal chain structure. We consider the maximal chain structure $(P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{C})$ as an envelope of the chain structure $(P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{B})$ . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms $Aut(P, \mathfrak{G}_{1}, \mathfrak{G}_{2})$ , $Aut(P, \mathfrak{G}_{1}, \cup \, \mathfrak{G}_{2})$ , $Aut(P, {\mathfrak{C}})$ , $Aut(P, {\mathfrak{G}_{1}}, {\mathfrak{G}_{2}}, {\mathfrak{C}})$ related to a maximal chain structure $(P, {\mathfrak{G}_{1}}, {\mathfrak{G}_{2}}, {\mathfrak{C}})$ . The set $\mathfrak C$ of all chains can be turned in a group $(\mathfrak {C}, ·)$ such that the subgroup $\widehat{\mathfrak C}$ of $Sym \mathfrak{C}$ generated by ${\mathfrak{C}_{l}}$ the left-, by ${\mathfrak{C}_{r}}$ the right-translations and by ι the inverse map of $(\mathfrak{C}, ·)$ is isomorphic to $Aut(P, \mathfrak C)$ (cf. (2.14)).

Keywords:	51B20(2000)
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