As a layman, I suspect my question is ill-formed and I will do my best to explain what it is I mean.

I find the notion of algebraic structures absolutely fascinating. I understand that there exist structures that require certain properties of addition and multiplication to hold. As an example, a ring requires an abelian group with a second binary operation that is associative and distributes over the operation in the abelian group. A field is an example of another similar structure.

The Question

Given some set $A$ Is there a name for a structure $S$ that requires certain properties of $(A, +)$ and $(A, \cdot)$ to hold that also hold given $(A, \cdot)$ is the additive structure and $(A, +)$ the multiplicative structure? I am trying to say that it matters not which structure we consider as additive and multiplicative as the properties of the $S$ hold regardless. In this sense $(A, +)$ and $(A, \cdot)$ are dual to eachother. Do these objects exist and are they interesting to study?

My gut feeling says that I am failing to write my intent accurately and apologize in advance.

  • $\begingroup$ Are you referring to, for example, the fact that the real numbers $\mathbb R$ with respect to addition is isomorphic as groups to the positive real numbers $\mathbb R^+$ with respect to multiplication, the isomorphism being $x\mapsto e^x$. $\endgroup$ – Gregory Grant Apr 29 '15 at 14:38
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    $\begingroup$ Are you asking: does there exist a set $A$ and two binary operations $\oslash$ and $\odot$ so that $(A,\oslash,\odot)$ and $(A, \odot, \oslash)$ produce the same algebraic structure? $\endgroup$ – Sloan Apr 29 '15 at 14:40
  • $\begingroup$ This is what I am asking Sloan. I think i could find some trivial examples however I wonder if there are interesting examples. $\endgroup$ – nSheahan Apr 29 '15 at 14:41
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    $\begingroup$ In particular, boolean algebras maybe? $(B,\vee,\wedge)\cong(B,\wedge,\vee)$ $\endgroup$ – Alexey Burdin Apr 29 '15 at 14:52

As I understand it, you're looking for a set $A$ with two operations $\cdot$ and $+$ such that $(R, +, \cdot)$ is a ring and $(R, \cdot, +)$ is a ring.

Unfortunately, this can't really happen. It's not hard to show that in any ring, $0 \cdot r = 0$ for all $r \in R$. So $1 + r = 1$ for all $r \in R$ also, since $(R, \cdot, +)$ is a ring. But this implies $r = 0$ for all $r \in R$, so the only example is the trivial ring.

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  • $\begingroup$ Thank you for your answer. This is interesting as it relates to rings but I am wondering what happens if we remove the requirements that the structure is a ring. Are there any interesting examples of structures that have $(A, +, \cdot) = (A, \cdot, +)$ in general, not necessarily a ring? $\endgroup$ – nSheahan Apr 29 '15 at 14:47

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