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We are all familiar with the concept of prime numbers from a young age. Numbers that are divisible only by 1 and itself, or that have no factors other than 1 and itself. Over time, mathematicians developed abstract algebra and ring theory and found ways to generalize primes even further. But rings still involve addition and multiplication, and the definition is still based on multiplication. But can't the concept of primeness be thought of differently? And how can the concept of primality be further generalized?

Let's remember Schubert's theorem as an example. In the knot theory of topology there are so-called prime knots, i.e. mathematical objects. So do you think the concept of prime here and the concept of prime in numbers are completely opposite? I don't think so, there is no reason not to think that there is a deep connection between them.

Theorem: Horst Schubert (1919-2001) every knot can be uniquely expressed as a connected sum of prime knots.

Theorem (Fundamental Theorem of Arithmetic): every integer greater than 1 can be represented uniquely as a product of prime numbers

Note the multiplication and connected sum operation here.

I suspect that there is a much more general concept of prime in mathematics to discuss.

rschwieb
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Egehan Eren
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  • A good first start could be to look into prime ideals, which are the direct generalisation of primes in the setting of commutative algebra, especially since $p\mathbb{Z}$ is a prime ideal of $\mathbb{Z}$ iff $p$ is prime in the usual sense. Otherwise I don't really think I could flesh out an answer myself. – Bruno B Dec 12 '24 at 16:43
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    Technically, it does apply to addition. The the natural numbers, there is only one additive prime, $1.$ In the integers, there are no additive primes, because every integer has an (additive) inverse. – Thomas Andrews Dec 12 '24 at 16:46
  • What does "connected sum" mean? To my mind a related question should be: can the concept of "connected sum/addition" be considered to be equivalent to "product/multiplication". I'd think the answer is yes. – fleablood Dec 12 '24 at 16:47
  • But can't it be generalized to more other operations? – Egehan Eren Dec 12 '24 at 16:48
  • "But can't it be generalized to more other operations?" To binary group operations where not all elements have inverses, I'd imagine. – fleablood Dec 12 '24 at 16:50
  • Primes are numbers $ >1 $ that can be expressed only as a product of natural numbers that are themselves and $1$. All natural numbers $>1$ can be expressed as a sum of any natural number less than them and another natural number, so the additive analogue is not interesting – J. W. Tanner Dec 12 '24 at 16:51
  • I know that the additive analogue is not interesting, what I mean is to generalize the notion of primality to other operations. This is not necessarily addition. – Egehan Eren Dec 12 '24 at 16:52
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    Prime and irreducible elements are defined in every commutative monoid. That's standard and all your examples fall under this notion. Does this answer your question? If not, please clarify. – Martin Brandenburg Dec 12 '24 at 17:21
  • The notion of primality extends to dynamical systems and ergodic theory. A transformation is prime if it has no non-trivial factors. A factor is determined by an invariant sub sigma algebra. There is always a trivial factor using the algebra based on the whole space $X$ and the empty set $\emptyset$. A rotation on the torus is not prime, since it can be expressed as the cross product of two rotations on the circle: $T_{\alpha}xT_{\beta}$. It is not easy to identify prime transformations. Here's a paper that proves a particular example is prime: https://tinyurl.com/yvse57fp. – The Other Terry Dec 12 '24 at 18:03
  • Elaborating on @Martin's comment, in fact much of divisibility theory in integral domains is a special case of that for monoids, see divisibility groups. – Bill Dubuque Dec 12 '24 at 18:06
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    There are a lot of results in mathematics that state in various ways that all objects of a certain type can be expressed as a certain type of "joining together" of "fundamental objects", e.g. classification theorems. @user65023 has mentioned one such example. Another are some results involving linear orders (and quasi-orders) -- see this MSE answer by Andrés E. Caicedo. Still another is the classification of $2$-surfaces with connected sum as the "joining together" operation. – Dave L. Renfro Dec 13 '24 at 16:34
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    For connected sum of topological manifolds see this MathOverflow post. – Lee Mosher Jan 01 '25 at 16:47

1 Answers1

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Why does the concept of prime, primality only apply to multiplication and division?

One does not have to assume that they do...

Any discussion about what primes are should also mention what irreducibles are and what atoms are. They often get lumped together because for things like the integers, they amount to the same thing.

All three conditions have roots in lattice theory, and so you would probably be interested in this wiki.

There, in a lattice (or a join semilattice, at least),

$x$ is prime if $x\leq y\vee z$ implies $x\leq y$ or $x\leq z$;

$x$ is irreducible if $x=y\vee z$ implies $x=y$ or $x=z$; and

$x$ is an atom if the lattice has a least element $0$ and there is no $y$ other than $0$ and $x$ such that $0\leq y\leq x$.

Notice there are two ingredients: the partial order and the join operation. The partial order determines atoms (if they exist) and the join helps assemble them into other elements. The existence of atoms, and whether or not they generate everything in the lattice, and uniqueness of representation are all fundamental questions for these topics.

That's why you see these definitions on the same wiki page:

A lattice or is called atomic if for every $x\neq 0$, there's an atom $a\leq x$; and atomistic if $x$ is the supremum of atoms less than $x$.

The comments bring up that commutative monoids are relevant. How do they fit in? Well, with the commutative binary operation in a monoid you get an upper semilattice of principal ideals of the monoid where $(a)\leq (b)$ means $(b)\subseteq (a)$. It can be checked that it has least element $(1)$ and that $(a)\vee (b)=(a)(b)=(ab)$ works as a least upper bound.

Compare that to the definitions in ring theory used with domains:

$x$ is prime if $x|yz$ implies $x|y$ or $x|z$;

$x$ is irreducible if $x=yz$ implies $x=uy$ or $x=uz$ for some unit $u$.

(In integral domains) it's this passage from elements to principal ideals generated by elements that precipitates rephrasing $(a)=(b)$ as an equivalence relation between elements in terms of the elements and units. For a general commutative monoid you could very well just say "$a\cong b$ if $(a)=(b)$" instead.

The lattice-theoretic notions above all become classical topics in commutative algebra for factorization questions: atomic domains, factorial domains, and best of all unique factorization domains, like $\mathbb Z$.

And also, for your example of knot theory, I read that the operation of summing knots in 3-space creates a commutative monoid, giving rise to its own sort of lattice. I am not sure what the situation is in general... I am not knowledgeable about knot theory.

rschwieb
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