I'm working with an idempotent semiring which have families $C^v, \hat{C^v}$ of elements with the following properties:
$$ {C}^v_i \hat{C_i^v} = 1 $$ $$ \sum_i \hat{C_i^v} {C}^v_i = 1 $$ $$ {C}^v_i \hat{C_j^v} = 0 \quad\text{where}\quad i \neq j $$ $$ {C}^v_i \hat{C_j^w} = \hat{C_j^w} {C}^v_i \quad\text{where}\quad v \neq w $$
As we see $\hat{C_i^v}$ is close but not quite an inverse of $\hat{C_i^v}$.
Also the last property often does not fully hold in many cases.
Have such or similar structures appeared or have been explored elsewhere?
Do any of these properties have a name that I could google?
Thanks for any and all leads!