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

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)).