HIGHER-ORDER BERNSTEIN ALGEBRAS GIVEN BY SYMMETRICAL BILINEAR-FORMS

Let (A, omega) be a kth-order Bernstein algebra and let N be the kerne l of omega. This article studies the structure of such algebras in whi ch N-2 has dimension one. The algebras are of two types, I and II, acc ording 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 assoc iative kth-order Bernstein algebras with dim N-2 = 1 are then consider ed: they turn out to be Bernstein algebras of at most second order, an d multiplication tables for these algebras over the real field are giv en. Finally, second-order Bernstein algebras of type II are examined a nd a structure theorem for them is obtained. (C) Elsevier Science Inc. , 1997