Genetic algebra

Remarks

Several closely related definitions are in use; this one asks for a basis in which the multiplication is triangular. The linear map $w$ with $w(c_0) = 1$ and $w(c_i) = 0$ for $i \ge 1$ is then a weight, so $(A, w)$ is a baric algebra. Indeed, the conditions force $c_0 c_0$ to have weight $1$ and all other products of basis vectors to have weight $0$.

Example: the gametic algebra with basis $A_1, A_2$ over $\Q$, $A_1^2 = A_1$, $A_2^2 = A_2$, $A_1 A_2 = \tfrac12(A_1 + A_2)$. Take $c_0 = A_1$ and $c_1 = A_1 - A_2$. Then $c_0^2 = c_0$, $c_0 c_1 = A_1 - \tfrac12(A_1 + A_2) = \tfrac12 c_1$, and $c_1^2 = A_1 + A_2 - (A_1 + A_2) = 0$, so all three clauses hold.

Non-example: $K^2$ with the componentwise product is not genetic. For a basis as above, $w$ is a weight and clause 3 gives $c_1^2 = 0$ for the vector $c_1$ that spans $\ker w$. But the only weights on $K^2$ are the two coordinate projections, and the kernel of each is spanned by an idempotent $e$, with $e^2 = e \ne 0$. So $c_1 = t e$ with $t \ne 0$ and $c_1^2 = t^2 e \ne 0$, a contradiction.