# Kleene algebra without right distributivity?

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 ?

Alex

• I would call it something like 'left Kleene algebra'. But it doesn't really help in searching.. Can you share us some interesting features/properties/usage of these? – Berci Feb 19 '13 at 11:29
• Well, I am formalizing a description language that mimics the structure of CFGs, but applied on an typed process space. My "alphabet" is a set of processes, the natural multiplication '.' is process composition, and the addition '+' is the parallelization of 2 processes. We also want to be able to "join" the results from two parallel processes if needed, and thus our processes are typed, and (a + b) is a process that outputs a Pair of results. Given a process 'c', we can maybe type (a + b).c or (a.c + b.c), but not both, which yields the contradiction to the right-distributive axiom. – Alex Repain Feb 19 '13 at 13:47
• Also, I found mentions of "left-handed Kleene algebra", but the 'left-handed' part refers to the * axioms of the algebra, and not the distributivity in the semiring... As for the properties of the structure, if it already has been studied, that is what I am after ! – Alex Repain Feb 19 '13 at 13:50