Geometry of kinematic \(K\)-loops
A loop \((L,+)\) is called \(K\)-loop, if \(\forall a,b\in L\), \(\delta_{a,b}:=((a+b)^+)^{-1}\circ a^+\circ b^+\) (where \(a^+:L\to L\); \(x\to a+x)\) is an automorphism of \((L,+)\), \(\delta_{a,b}=\delta_{a,b+a}\) and \(-(a+b)=-a+(-b)\) (where \(-a\in L\) is defined by \(a+(-a)=0)\). The authors call a \(K\)-loop kinematic if \(\forall a,x,y\in L\setminus\{0\}\), \(\delta_{a,x}=\delta_{a,y}=id\) implies \(\delta_{x,y}=id\), or equivalently if \(\{[a]\setminus\{0\}\mid a\in L\setminus\{0\}\}\), (where \([a]:=\{x\in L\mid\delta_{a,x}=id\})\) forms a partition of \(L\setminus\{0\}\). To such loops the authors associate a right and left geometry \((L,{\mathcal G})\) and \((L,{\mathcal L})\) by setting \({\mathcal G}:=\{[a]+x\mid a,x\in L,\;a\neq 0\}\) and \({\mathcal L}:=\{x+[a]\mid a,x\in L,\;a\neq 0\}\) and defining: \([a]+x|[b]+y:\Leftrightarrow[a]=[b]\). They prove: Each \([a]\) is a commutative subgroup of \((L,+)\); \((L,{\mathcal G},|)\) resp. \((L,{\mathcal L})\) are incidence spaces with parallelism resp. with \(\{a^+\mid a\in L\}\subset\text{Aut}(L,{\mathcal L})\); and ``\(x+[a]|y+[b]:\Leftrightarrow[a]=[b]\)'' defines a parallelism for \((L,{\mathcal L})\) iff \(\forall a,x\in L\), \(\delta_a,\delta_{a,x}(a)=id\). Finally they present several examples where the automorphisms \(\delta_{a,b}\) are either the identity or fixed point free.
scientific article; zbMATH DE number 878376