7

I wonder what this stunning formal analogy between the definitions of being co-prime (for two integers) and being prime (for one integer) might reveal – and how:

$\alpha, \beta$ are co-prime iff

$$(\forall x)\ \alpha|x \wedge \beta|x \leftrightarrow \alpha\beta|x$$

$\alpha$ is prime iff

$$(\forall xy)\ \alpha|x \vee \alpha|y \leftrightarrow \alpha|xy$$

Note that and how the two definitions are equivalent modulo swapping

  • $\wedge$ and $\vee$ on the left side

  • constants and variables on the right side together with

  • the direction of divisibility $|$ with respect to the product [thanks to user Wojowu]

  • the direction of inference $\rightarrow$ [thanks to user Roll up and smoke Adjoint]

Hans-Peter Stricker
  • 18,719
  • 1
    Note you also swap the direction of divisibility. This is an interesting observation, but I don't think there is any "stunning" meaning behind it. – Wojowu Nov 02 '18 at 19:28
  • Note that you can also replace $\rightarrow$ with $\leftrightarrow$ since each of those propositions on either side are in isomorphism with one another. – Daniel Donnelly Nov 03 '18 at 03:29
  • There is most likely a nice interpretation in the language of lattices/filters. – Andrew Dudzik Nov 03 '18 at 09:05
  • In the poset category for integers dividing one another $\text{Hom}(X,Y)$ consists of a single point or is empty. Let $(X \mid Y)$ denote the nonempty hom-set. Notice that multiplication by an element $\alpha \in \Bbb{Z}$ is a functor from the poset category to itself since $(X \mid Y) \implies (\alpha X \mid \alpha Y)$ and so on proves the functoriality of multiplication. – Daniel Donnelly Nov 04 '18 at 06:34

2 Answers2

1

In the poset category of integers dividing one another we write sometimes write $\alpha \to x$ instead of $\alpha \mid x$.

The "product" $\alpha \beta$ of two coprime objects $\alpha, \beta$ is such that $\alpha \to \alpha \beta \leftarrow \beta$ and for any object $x$ such that $\alpha \to x \leftarrow \beta$ then there is a unique arrow $\alpha \beta \to x$. That is precisely the definition of coproduct in a general category.

Thus when $\alpha, \beta$ are coprime, then the category definitely has a coproduct for them.

Thus $\wedge$ is encoded in the fact that $\alpha \to x \leftarrow \beta$, ie. both morphisms exist simultaneously. I believe product is $\gcd(\alpha, \beta)$.

Actually, it turns out that the coproduct exists for any two integers and it's $\text{lcm}(\alpha, \beta)$.

Primality is difficult because of the $\vee$. So create a new definition. An arrow into a coproduct is prime when there exists a morphism into at least one of the coproduct's components such tha the relevant triangle commutes.

Notice we used coproduct here which is $\text{lcm}(\alpha, \beta)$ since the definition of prime is equivalent to $p \mid \text{lcm}(\alpha, \beta) \implies p \mid \alpha \vee p \mid \beta$.

So take the contrapositive of that. $p \nmid \alpha \wedge p \nmid \beta \implies p \nmid \text{lcm}(\alpha, \beta)$.

Form the "negated" poset category of integers not dividing one another. It's formed by mapping each hom-set to $\varnothing$ when it's nonempty and vise-versa, so it's not a functor from the original category.

Daniel Donnelly
  • 22,288
0

The second can become:$$(\forall xy)\alpha\mid xy\to\alpha\mid x\lor\alpha\mid y$$ The first is: $$(\forall x)\alpha\beta\nmid x \to \alpha\nmid x \lor \beta\nmid x \lor \alpha\beta \gt x\lor(\alpha,\beta)^2\nmid x$$ Not sure it reveals much though.