Partial Steiner triple systems as bases of flexible algebras (Q2717066)

An algebra \(V\) is flexible if \((xy)x=x(yx)\), for all \(x,y\in V\). A basis \(B\) of \(V\) is called a Steiner basis if \(B\cup(-B)\cup\{0\}\) is closed under the algebra multiplication of \(V\), if either \(xy=yx\) or \(xy=-yx\) (excluding characteristic 2 here), and if for every \(x,y,\in B\), whenever \(xy\neq yx\) and \(xy\in B\), then \((xy)y\neq y(xy)\in\{x,-x\}\) and \(x(xy)\neq (xy)x\in \{y,-y\}\). Such a basis defines a partial Steiner triple system on \(B\) in the obvious way (2 elements are contained in at most one triple \(\{x,y,z\}\) with \(z=xy=-yx\)). The author proves a lot of small properties of flexible algebras (with and without a Steiner basis), and constructs such an algebra from any given abstract partial Steiner triple system. NEWLINENEWLINEThe motivation for such work is a bit unclear to me, but it certainly generalizes the representation of the 7 standard basis elements (except 1) of the classical Cayley division algebra over the real numbers using the lines of the projective plane of order 2 (the so-called Fano configuration) as noticed by \textit{H. Freudenthal} [Geom. Dedicata 19, 7--63 (1985; Zbl 0573.51004)].