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Is there a name for a theory that has all axioms of a semiring but an axiom that mandates multiplicative identity?

vonbrand
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google
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    Following one naming convention, we should call such a thing a "rg". I suppose the convention has broken down at this point.... –  Apr 12 '13 at 13:54

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Semiring theory is plagued by a mess of vocabulary the remains unresolved. Still, there are two books I always check when dealing with them.

In Golan's Semirings and Their Applications, a semiring without identity is called a hemiring.

A newer, also fairly comprehensive book on semirings is Graphs, Dioids and Semirings: New Models and Algorithms by Gondran and Minoux. They list semirings without identity as pre-semirings.

Hemiring in particular gets thousands of hits, although due to the mess of vocabulary in the theory, I can't guarantee what percentage of those is what you seek.

While I don't think those terms are going to be universally used, I think the references in both these books would help you track down specific information about such structures.

rschwieb
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  • Ah. Isn't it a bit weird to invent new names in each case which you cannot remember instead of following a general principle which "generates" the terminology? – Martin Brandenburg Apr 12 '13 at 15:32
  • @MartinBrandenburg Yeah, you have to admire any author trying to write a book on the topic, since they are forced to distill the orgy of random terminology invented by computer scientists into something meaningful. Even after distillation there are too many wacky names... – rschwieb Apr 12 '13 at 16:25
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This would be the fusion of semirings and pseudo-rings (or rngs), so they should be called pseudo-semirings (or semirngs). But with google I couldn't find any usage for this (only with other meanings).

Martin Brandenburg
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