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,


  • $\begingroup$ 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? $\endgroup$ – Berci Feb 19 '13 at 11:29
  • $\begingroup$ 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. $\endgroup$ – Alex Repain Feb 19 '13 at 13:47
  • $\begingroup$ 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 ! $\endgroup$ – Alex Repain Feb 19 '13 at 13:50

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!

  • $\begingroup$ Thanks a lot for the pointers, Graphs, Dioids and and Semirings seems indeed to circumvent my field and explore it pretty extensively ; I'll have a peek at it! $\endgroup$ – Alex Repain Feb 20 '13 at 9:17

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