Ahoy!

@MaliceVidrine Hi.

@IDUTDP Hello!

Good afternoon.

:)

May i ask you a question?

I just don't think it's worth a "proper" question . . .

Go for it.

I have at least two structures. I know there's at least one homomorphism between them, problem is: i don't know anything about the properties of said homomorphism, or even how many they are . . .

So . . . i want to know about the properties of said homomorphism but don't know what to do . . .

Much less can construct even one . . .

Essentially they are order homomorphism between a partially ordered set with a particular algebraic structure and a totally ordered set with the same algebraic structure . . .

I think I need more details. What are the structures specifically? What category do they live in?

Algebraic structure, binary operation: clausurative, associative, commutative, cancellable, identity, only the identity has an inverse . . .

About the category . . . have no idea . . .

I don't know what "clausurative" or "cancellable" mean in this case.

I'm going to assume it's the category of preordered sets with this binary operation, and functions that preserve the order and the operation?

Cancellable: a+b=a+c --> b =c b+a=c+a --> b=c

yeah, not preorders though.

Full partial orders.

Ah, right, misread.

And I need to know not just the nature of the binary operation, but actually the definition of the two specific structures you're talking about. Knowing only that they have this kind of structure and that one is totally ordered isn't much to work with. Not knowing anything about the category of these objects generally, there's no category theoretic trick that's going to save us any work.

They could be the natural numbers as the totally order one . . .

And finite sets of some universe as the partially ordered one.

Also, bummer that . . .

Since it doesn't sound like the ordering operation and the addition interact (at least not that you've mentioned) you could always just look for monoid homomorphisms and then see if they preserve the ordering. But in general it sounds like you could easily end up with pairs of objects between which there are no homomorphisms.

How so?

because they don't preserve the algrebraic structure or because they don't even preserver the orderinf relation?

Because they need to preserve both and all it takes is making sure that if you preserve one, you don't preserve the other.

I think I have an example.

Please share!

That's progress for me!

Actually, not a counterexample. Probably just an explanation of why you have at least one silly morphism between these structures.

I was going to say take the natural numbers under their usual ordering and monoid operation; then take the structure with the natural numbers, their usual monoid operation, but the reverse ordering relation. But there's exactly one morphism between these structures: a homomorphism that maps everything to zero.

I.e. the zero morphism is always order preserving and a monoid homomorphism.

There we go.

(sorry, still waking up.)

It's ok.

And yeah, the trivial case.

I'll have to keep thinking, maybe something will come up .. .