Loops with Adjoints

Extract

It is shown in (6) how to represent certain sets of orthogonal Latin squares as a group together with a set of permutations of the group elements. The correspondence between 3-nets and loops is well known; for example, see (8). We shall consider a loop G together with a certain set of permutations on the elements of G and shall interpret such a structure as an incidence system in which the 3-net of the loop is embedded. Specifically, the permutations or “adjoints” will give rise to lines which may be adjoined to the 3-net of G in the sense of (3). The group of autotopisms of the loop determines a group of automorphisms of its 3-net analogous to collineations in an affine plane. We shall study the problem of extending these incidence preserving mappings to the adjoined lines. By analogy with the study of loops with operators, we shall consider homomorphisms of loops with adjoints and examine geometric consequences. Particular attention will be paid to the case where G has the inverse property and the adjoints are “linear.” The special case in which G is an abelian group is of geometric interest in that the corresponding incidence systems include the Veblen-Wedderburn affine planes.