3
$\begingroup$

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

Axioms:

  • $+$ is commutative and associative,

  • The multiplicative structure forms an abelian group

  • Multiplication distributes over addition.

Examples.

  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.
$\endgroup$
2
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ 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

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.