@Anindya

From [#46](https://forum.azimuthproject.org/discussion/comment/18027/#Comment_18027):

> point taken re "let's not fool anyone into thinking every monoidal preorder needs to have \$$\otimes\$$ be the join or meet" – one of the difficulties I have with monoidal categories is in grasping how we only care about the formal combinatory properties of the monoidal product, and not what it "actually is". So I tend to "reach" for familiar products, eg meet or cartesian or tensor, which can be misleading.

I don't know if you are a Haskell programmer, but [Data.Monoid](https://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Monoid.html) has a lot of canonical monoids. It has instances for [a], endomorphisms, First a and Last a. First a obeys \$$\forall x. x \neq I \implies x \otimes y = x\$$, like you sketch above (but it doesn't have the order you prescribe).

There are also some truly awesome machine learning examples in Izbicki's [*Algebraic classifiers* (2013)](http://proceedings.mlr.press/v28/izbicki13.pdf). Not only does Izbicki demonstrate how to model various popular machine learning algorithms as monoids, but he shows how to leverage monoidal (and group) structure for speeding up \$$k\$$-fold cross-validation from \$$\mathcal{O}(kn)\$$ to \$$\mathcal{O}(k + n)\$$ (given sufficient cloud computing resources).

From [#43](https://forum.azimuthproject.org/discussion/comment/18023/#Comment_18023):

> @Jonathan – I think isomorphisms are enough – if \$$Q\$$ is a preorder isomorphic to \$$P\$$, then we can transplant the monoidal structure on \$$P\$$ onto \$$Q\$$ using the same trick (convert the \$$Q\$$-elements into \$$P\$$-elements, multiply them in \$$P\$$, convert back again). The automorphism version (with \$$Q = P\$$) is just a special case of this.

I agree.

This is enough to prove an early hunch of mine in this thread which I later doubted.

*Every dense unbounded linear order can be equipped with a monoidal poset.*

(This is a sort of model theoretic answer to **Puzzle 63**.)

**Proof**.

In the case of \$$\aleph_0\$$ this is easy enough to see, since it's order-isomorphic by Cantor's theorem to \$$\mathbb{Q}\$$ so we can transfer the monoidal pre-order \$$\langle \mathbb{Q}, \leq, +, 0\rangle\$$ to it.

But by upward Löwenheim-Skolem, for any cardinality \$$\kappa\$$ there is a dense unbounded linear monoidal poset \$$\langle D, \leq, +_D, 0\rangle\$$ which is an [elementary extension](https://en.wikipedia.org/wiki/Elementary_equivalence#Elementary_substructures_and_elementary_extensions) of \$$\langle \mathbb{Q}, \leq, +, 0\rangle\$$ where \$$|D| = \kappa\$$.

Since the theory of dense linear orders is complete, then that means for any other dense linear poset with \$$\kappa\$$ many elements it must be order isomorphic to \$$\langle D, \leq, +_D, 0\rangle\$$ by some isomorphism \$$\phi\$$. Then we can use Jonathan's idea in [#41](https://forum.azimuthproject.org/discussion/comment/17997/#Comment_17997) along with \$$\phi\$$ to embed \$$+_D\$$.

\$$\Box\$$