Higher-order Bernstein algebras given by symmetric bilinear forms

Let (A, omega) be a kth-order Bernstein algebra and let N be the kernel of omega. This article studies the structure of such algebras in which N-2 has dimension one. The algebras are of two types, I and II, according as N-2 subset of or equal to U or N-2 not subset of or equal to U. A characterization of the algebras of type I is given. Power associative kth-order Bernstein algebras with dim N-2 = 1 are then considered: they turn out to be Bernstein algebras of at most second order, and multiplication tables for these algebras over the real field are given. Finally, second-order Bernstein algebras of type II are examined and a structure theorem for them is obtained. (C) Elsevier Science Inc., 1997