I'm facing a mathematical structure that has everything of a Kleene algebra (S, +, ., 0, 1, *), except that the multiplication '.' is not right-distributive over the addition '+'.
I reckon that it could be defined as a weakened version of Kleene algebra, where the semiring (S, +, ., 0, 1) is weakened to a (left) near-semiring, but I haven't found that description used anywhere until now.
Is there a known name for such a structure, that could hint me to some literature ?
Thanks by advance,