As in the title, I am looking for the right name for this algebraic structure, which is exactly as an idempotent semiring, apart from the fact that multiplication does not right-distribute over addition.

The name I would imagine is something like "idempotent left semiring", since multiplication still left-distributes over addition, but I'm unable to locate a proper reference in the literature.

To make things clear, by semiring I mean an algebraic structure in which there is an associative and commutative additive operator ("+") as well as an associative multiplicative operator ("*"), which is both left- and right-distributive over addition. An idempotent semiring is a semiring in which both operators are idempotent.

  • $\begingroup$ Are there examples of these structures? $\endgroup$ – Joppy Oct 11 '17 at 12:22
  • $\begingroup$ Crosspost. Please avoid crossposting simultanously, and if you do for some reason, be transparent on both ends about it. $\endgroup$ – rschwieb Oct 11 '17 at 13:15
  • $\begingroup$ My bad. I mentioned the crosspost on the other forum, but I forgot to do it here. Apologies $\endgroup$ – Maiaux Oct 11 '17 at 13:22

They are called idempotent near-semirings, but I disagree with your definition of an idempotent semiring. An idempotent semiring is a semiring in which the addition is idempotent. See also Semiring on wikipedia.

  • $\begingroup$ The Wikipedia entry for near-semirings actually uses right distributivity in its definition but says that + and * are related by one (right or left) distributive law. How do I distinguish between these two cases? Should I add "left" to the name of the structure, so that it becomes "idempotent left near-semiring"? $\endgroup$ – Maiaux Oct 11 '17 at 13:10
  • $\begingroup$ @Maiaux That sounds sensible. It's not like terminology police are going to come arrest you for being clear by adding "left". $\endgroup$ – rschwieb Oct 11 '17 at 13:16

I've transferred my answer on matheoverflow over here: I did not realize it was crossposted, and it is a better question for this forum.

I'm pretty sure you are looking for a near-semiring. You could call it an "idempotent near-semiring" using left-right if necessary.

There might be other/more terms suggested in Gondran and Minoux's book:

Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.

I think they call a commutative semigroup equipped with another semigroup operation that is left distributive a left pre-semiring. They do not even require an absorbing element or identity for the second operation. One could tack on "idempotent" to this as well.

I can't really vouch for the impact of this book on terminology. But, terms for ring-like structures this general are a bit of a mess, and I can't complain much about an attempt to outline some sort of consistent system in a single book.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.