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Let $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ denote the non-negative real numbers with usual addition and usual multiplication. Obviously, this is not a field, because $0$ is the only additively invertible element.

What kind of algebraic structure does $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ consitute?


Update:

As @Mathmo123 wrote, it is a semiring. However, the semiring axioms do not take into account that all non-zero elements are multiplicatively invertible (like in a field).

I am interested in $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$, because it plays an important role in many contexts. For example, it is the image of the maps $x \mapsto x^2$ and $x \mapsto |x|$. To refer to the images as a semiring with non-zero multiplicative inverses is strange to me. There should be a better term for this type of (obviously relevant) algebraic structure.

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    $\begingroup$ It would be called a Semiring or a "Rig" $\endgroup$ – Mathmo123 Feb 9 '16 at 21:11
  • $\begingroup$ @Mathmo123: Oh, I immedately see this. But if we take into account that, in addition to the semirng axioms, all non-zero elements have multiplicative inverses, can we say more about the structure? $\endgroup$ – Björn Friedrich Feb 9 '16 at 21:22
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    $\begingroup$ While $\mathbb{R}_{\geq 0}$ does occur in many contexts, I find your examples to be funny, as they don't preserve the additive structure you're including in your description of this object. It might be more natural just to think of $\mathbb{R}_{\geq 0}$ as a monoid (with $0$) under multiplication. $\endgroup$ – Stahl Feb 9 '16 at 22:02
  • $\begingroup$ @Stahl: Can you elaborate on this, please? The two maps are no homomorphisms, right. Nonetheless, their image is $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$. For example, $|x| + |y|$ and $|x| \cdot |x|$, but also $\frac{|x|}{|y|}$ for $|y| \neq 0$, are valid operations. $\endgroup$ – Björn Friedrich Feb 9 '16 at 22:15
  • $\begingroup$ If you want to give a name to some structure, you will/should probably be concerned in other objects that have this structure and in maps between such objects that preserve all the structure you've given the objects (of course, there are notable exceptions). The maps you've described from $\mathbb{R}\to\mathbb{R}_{\geq 0}$ preserve the multiplicative structure, but not the additive structure, and it would make more sense to say that they are maps of monoids (preserving the multiplication) than to say they're maps of structures with a $+$, although you can add on both. $\endgroup$ – Stahl Feb 9 '16 at 22:29
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"Semifield" seems like one good choice (although it appears to have another meaning as well - see https://en.wikipedia.org/wiki/Semifield). For maximum clarity, I'd probably say "division semiring" (or "ordered division semiring" if I wanted to emphasize the ordering).

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