The Bernstein problem in dimension 5
A Bernstein algebra \(A\) has a Peirce decomposition relative to an idempotent \(e\), given by \(A= \text{Re} (+)U(+)V\). If \(UV+V^2=0\), the algebra is called ``regular''. If \(U^2=0\) the algebra is called ``exceptional''. The authors study the Bernstein algebras with are not regular and not exceptional and are of dimension 5. They first show that Bernstein algebras are not regular and not exceptional if and only if they have a basis \(\{e,u1,u2,v1,v2\}\) and one of six types of multiplication tables. They then classify stochastic not regular not exceptional Bernstein algebras of dimension 5. They give two types of these algebras and show that all such algebras are of one of these two types.
scientific article; zbMATH DE number 827371