The Number of Structures of Finite Relations

1. Preliminaries. Let A be a dyadic relation defined on a finite set of n elements. It is convenient to regard A as an nXn matrix over the two-element Boolean algebra, so that a,, = l if and only if iAj. There is then an obvious correspondence of relational concepts [l; 2]2 with matrix notions, interpreting conjunction as "coordinatewise" multiplication, relative product as matrix product, etc. In this paper, however, matrix notation and Boolean l's and O's are adopted merely for convenient reference. If A and B are two relations defined over the set N= {l, • ■ ■ , n}, then they are isomorphic just when there is a permutation tt ol N such that A has the same matrix with respect to N as B with respect to w(N) [cf. 3, *151.0l]. Formally, for every relation A defined on N and every t in ©„ (the symmetric group of n letters), define a transformation tT of A by