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 '+'.
http://en.wikipedia.org/wiki/Kleene_algebra
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.
http://en.wikipedia.org/wiki/Near-semiring
Is there a known name for such a structure, that could hint me to some literature ?
Thanks by advance,
Alex
Once I ran across Graphs, Dioids and and Semirings by Gondran and Minoux, and discovered it was a pretty well written book about such things.
I can't be positive it has exactly what you mention, but I remember it contained very detailed nomenclature for organizing generalizations of rings (for example: single-sided axioms, exactly as you are looking for). Since such generalizations have exploded in the past 40 years, and everybody chooses different names for stuff, the authors had a tough task of shoehorning everybody's terminology into a single comprehensible book. It is also pretty new, so I'd start there!
Another one I enjoyed was Golan's Semirings and their applications; however, it is much older, and I have even less confidence that it addresses exactly what you are interested in.
Good luck!