A ternary algebra arising from association schemes on triples
In a previous paper [J. Comb. Theory, Ser. A 55, 204-234 (1990; Zbl 0761.05098)], the authors considered a ternary analogue of association schemes involving ternary (instead of binary) relations and proved that the 3-dimensional adjacency matrices of the relations of the scheme form a ternary algebra under a suitable composition. This may be considered as the ternary analogue of the well-known Bose-Mesner algebra of an ordinary association scheme. Examples were obtained from the groups \(\text{PSL} (2,q)\), \(\text{AGL} (1,q)\) and from 2-designs. In the paper under review, the authors study ternary algebras formed by three-dimensional arrays in general, with particular emphasis on so-called identity pairs and inverse pairs. Some more concrete examples are also discussed in detail.
scientific article; zbMATH DE number 579541