A slight generalization of affine extension

Stream: Equational

[Bruno Le Floch]

From a cursory look, the reference THE STRUCTURE OF MEDIAL QUANDLES that @Fox Room listed seems interesting more broadly: they seem to use piecewise linear models, which we haven't explored much.

[Fox Room]

It's a very neat paper that I have by no means taken the time to absorb the full breadth of.

[Bruno Le Floch]

It seems their construction is slightly more general than the affine extensions we have used a lot. An affine extension of a base magma $G$ with product $\circ$ is a product set $M=G\times F$ with $(x,s)\times(y,t) = (x\circ y, a_{x,y} s + b_{x,y} t + c_{x,y})$ where $s,t\in F$ an abelian group and the coefficients are allowed to depend on $x,y$. In their setting they also allow the fiber $F$ to depend on $x$. So for instance $a_{x,y}$ is no longer an endomorphism, but a linear map from $F_x$ to $F_{x\circ y}$.

[Matthew Bolan]

I actually have encountered these before. They are very natural since if $C_E$ is the category of $E$-magmas, then linear magmas are just abelian objects in $C_E$, and the magmas you have just described are abelian objects in the slice category $C_E/G$.

[Bruno Le Floch]

I wonder if this circumvents some of the no-go theorems we had for the finite 677→255 implication.

[Terence Tao]

I believe the no-go theorem for linear extensions in Lemma 13.4 of the blueprint https://teorth.github.io/equational_theories/blueprint/677-chapter.html still works when the fiber depends on the base point.