Question. Is there an accepted name for algebraic structures that, like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$, are models of the algebraic theory presented as follows?

Sorts: $U$

Functions: $$+ : U \times U \rightarrow U$$ $$1 : U, \qquad \times : U \times U \rightarrow U, \qquad x \mapsto x^{-1} : U \rightarrow U$$


  • $+$ is commutative and associative,

  • The multiplicative structure forms an abelian group

  • Multiplication distributes over addition.


  1. $\mathbb{Q}_{>0}$ (which is the initial such algebra)
  2. $\mathbb{R}_{>0}$
  3. $\mathbb{Z},$ with the symbol $\times$ interpreted as $+_\mathbb{Z}$, the symbol $1$ interpreted as the element $0_\mathbb{Z}$, the notation $x \mapsto x^{-1}$ interpreted as $x \mapsto -x$, and the symbol $+$ interpreted as either max or min.

That's a semfield. An additive identity might be additionally required.

  • $\begingroup$ Is there a less ambiguous name? Planet math gives no fewer than three possible (non-equivalent) meanings for the term "semifield," none of which are precisely the concept I"m looking for; in particular, it is vitally important to me that $0$ isn't included in the signature, otherwise we don't get an algebraic theory. Note also note that $\mathbb{Q}_{\geq 0}$ is not a model of the theory I'm interested in, since $0 \in \mathbb{Q}_{\geq 0}$ does not have a multiplicative inverse. $\endgroup$ – goblin GONE Dec 26 '14 at 17:24
  • $\begingroup$ I don't think there is one. It's a little studied area. You should definitely take a look at Golan's book on semirings, but I doubt you will find exactly what you're looking for there. Also, it's not that great a book in my opinion, but the only one anywhere near comprehensive to my knowledge. I'm not an expert (I'm hardly a mathematician), so perhaps you will find what you're looking for if you dig through papers referenced there. I haven't, but it's certainly something you should take a look at if you want to know about anything that resembles a semiring. $\endgroup$ – Bartek Dec 26 '14 at 19:04
  • $\begingroup$ Okay, thanks for the reference. $\endgroup$ – goblin GONE Dec 26 '14 at 19:05
  • $\begingroup$ Well, perhaps "little studied" is too much of a word. There's plenty of literature on semirings. It's just that they don't seem to admit much of a sensible theory. And if you look at the book you might think it's a bit chaotic for exactly that reason. $\endgroup$ – Bartek Dec 26 '14 at 19:09

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