Second order Bernstein algebras of dimension \(4\) (Q1906785)

A Bernstein algebra \((A,w)\) over a field \(K\) is a commutative algebra with a nonzero algebra homomorphism \(w:A\to K\) that satisfies \((xx)(xx) = w(x)^2xx\) for every \(x\) in \(A\). A second order Bernstein does not satisfy the Bernstein identity, but does satisfies \(((xx) (xx))^2=W(x)^4(xx)(xx)\). Second order Bernstein algebras \(A\) have the Peirce decomposition \(A=Ke(+) U(+)V\) where \(eu=1/2 u\) and \(e(ev)=0\). Let \(C=\{v \mid ev=0\}\). The study of these algebras is interesting because different idempotents give fundamentally different decompositions. The algebra is called ``coherent'' if for every idempotent \(Ce\neq Ve\); ``noncoherent'' if for every idempotent \(Ce=Ve\); otherwise, it is said to have ``double presentation''. The paper is a lengthy, but well organized case by case consideration of all the possibilities. Some of the these cases are based on the behavior of \(A^2\). The algebra is called ``nuclear'', if \(A^2=A\); ``2-nuclear'', if \(A^2\) is a nuclear Bernstein algebra; ``2-exclusive'', if \(A^2\) is a Bernstein algebra and \(U^2=0\). The major theorem states: If \(A\) is a 2-exclusive second order Bernstein algebra of type (2,2), then \(A\) has a standard basis \(\{e,u,v1,v2\}\) with \(ee=e\), \(eu=1/2 u\), \(uu=ev1=0\) and the other nonzero products belong to one and only one of the types listed below. There are 70 of these types, which are distributed over five cases of noncoherent algebras, six cases of coherent algebras, and seven cases for algebras with double presentation.