The defect in an invariant reflection structure
The pair \((P,I)\) is called an \textit{involution set} if \(P\neq\emptyset\) is a set and \(I\subseteq J := \{\sigma\in SymP \mid \sigma^2 = id \}\). For \(a, b \in P\) and \(\alpha,\beta \in \langle I\rangle\), let \(\widetilde{a,b} := \{\iota \in I \mid \iota (a)=b \}\) and \(\tilde{a} :=\widetilde {a, a}\). An involution set \((P,I)\) is called a \textit{reflection structure} if it is regular, i.e. if \(\forall a,b \in P:| \widetilde{a,b} | = 1\), and \textit{invariant} if \(\forall\alpha \in I : \alpha\circ I \circ \alpha = I\). For an invariant reflection structure \((P,I)\) the authors define the following two \textit{defect functions}, \[ \rho: P^3 \rightarrow SymP; (a,b,c) \mapsto \rho_{a;b,c} := \widetilde{a,c} \circ \widetilde{b,c} \circ \widetilde{a,b} \] and \[ \delta : P^3 \rightarrow SymP; (a,b,c) \mapsto \delta_{a;b,c} := \tilde{a} \circ \rho_{a;b,c}. \] If \(\delta\) is constant, then the reflection structure \((P,I)\) is called \textit{singular}; otherwise it will be called \textit{ordinary}. The authors show: if the invariant reflection structure \((P,I)\) is singular, then the set \(I \circ I\) := \(\{\alpha \circ \beta \mid \alpha, \beta \in I\}\) is a commutative regular subgroup of \(SymP\). They investigate invariant ordinary reflection structures \((P,I)\) in detail. First they introduce several three reflection axioms, then they define various blocks and determine relations between them. Let \(o \in P\), put \(P^\ast:=P\setminus\{o\}\) and let \((P,+)\) be the K-loop derived from \((P,I)\) in \(o\). The authors introduce two centralizers on \((P^\ast)^2\). They study the representation of blocks by centralizers and exchange conditions for centralizers. Finally they give necessary and sufficient conditions for some incidence structures to be linear spaces.
scientific article; zbMATH DE number 5963285