As the title implies, I am looking for the right name for an 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 find 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.

(I also posted this question on "Mathematics", but, on second thoughts, I feel mathoverflow is the right forum for it. Apologies if my assumption is wrong.)

  • $\begingroup$ No, math.stackexchange is the right forum for it. I imagine that most terminology questions belong there, since this forum is more for research-level questions. $\endgroup$ – rschwieb Oct 11 '17 at 13:14
  • $\begingroup$ How close is this structure to a lattice? Maybe you seek a semilattice with additional operators. Gerhard "Consider The Ups And Downs" Paseman, 2017.10.11. $\endgroup$ – Gerhard Paseman Oct 11 '17 at 14:40
  • $\begingroup$ @rschwieb I disagree. The research community may have greater experience and awareness of the research literature. (I had a question about terminology concerning "monoids in the category of Banach spaces and bounded linear maps", and someone here was able to point to some terminology in the literature) $\endgroup$ – Yemon Choi Oct 12 '17 at 0:19
  • $\begingroup$ @YemonChoi Ik, I'll keep it in mind in the future. I was going by the guidelines I've seen in the past on both sites, but since I not very active here, I haven't had a chance to come across nuances like this one. $\endgroup$ – rschwieb Oct 12 '17 at 10:50

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.


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