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Is there a name for the structure $(G,+,\cdot)$ where

  1. $(G,+)$ is a commutative monoid and
  2. left and right distributive laws hold?

($(G,\cdot)$ is not necessarily associative or commutative and need not have an identity.) It seems that such a structure may have an established name but I cannot find anything online or in my algebra books. It is not a semiring, nonassociative ring, or any other named structure that I can find. It would be great if there is a reference that I can look up and cite.

Thanks.

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  • $\begingroup$ @BillTrok Near-semirings only have one distribution law and also one absorption rule ($0\cdot a = 0$) $\endgroup$ – Stefan Perko Jun 5 '15 at 20:02
  • $\begingroup$ @StefanPerko whoops you are correct $\endgroup$ – Bill Trok Jun 5 '15 at 20:07
  • $\begingroup$ This book bestows many names upon many combinations of subsets of ring axioms. It may or may not appear in that book. Even if it does, the term used there may not be in common use. $\endgroup$ – rschwieb Jun 5 '15 at 20:37
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I've never heard of a name for said structure, but allow me to make one up:

A rig is a ring without additive inverses (it's missing negatives). So $(G,+)$ in this case is only required to be a commutative monoid. A rg (spoken as rug) is a rig without multiplicative identity (it's missing an identity). Now, your structure has to be a nonassociative rg.

EDIT: "According" to the book "Graphs, Dioids and Semirings", that rschwieb mentoined, your structure would be called something like "nonassociative pre-semiring with additive identity" or "nonassociative, nonunital, nonabsorbing semiring". I'm also making this up, partly, because the book does not actually define what you are looking for.

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