Finite Nets, I. Numerical Invariants

Introduction. A finite net N of degree &, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line {n ̂ 2) is simply a net of degree n + 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for 3 ^ k ^ n + 1, a set of k — 2 mutually orthogonal n X n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares. In the language of latin squares, the problem (explained in §1) of imbedding a net of N of degree kr order n in a net N r of degree k + 1, order n becomes the problem of finding a n w X ^ latin square orthogonal to each of k — 2 given mutually orthogonal n X n latin squares. Similarly, adjunction of a line corresponds to the determination of a common "transversal" (in the terminology of Euler [1]) to the k — 2 orthogonal squares. Further details of a historical nature will be found in the bibliography. On each finite net N we define an integer 4>(N), which may be regarded as an invariant in several ways. A necessary condition that a line can be adjoined to N is that <I>(N) = 1. (A necessary and sufficient condition is given in Theorem 1 (i).) We define a direct product Ni X N2 of nets Ni of the same degree and study the relation between <j>(Ni X N2) and the (j>(Ni) (Theorem 4). From these considerations we deduce the existence of nets of every order n to which no line can be adjoined (Theorem 5). Next we study the relation between the fis of homomorphic nets (Theorem 6) and we conclude the paper with an explicit evaluation of <j> for nets of degree 3 (Theorem 7).