One operation for generating algebraic structures - idea

Stream: Equational

[Michael Bucko]

Can an operation like this exist?

$(x \diamond y)^{ \diamond z} = x \diamond \big( (y \diamond (x \diamond z)) \diamond z \big)$

EDIT:

Another variant of this could be ($\phi$ could be (ml-wise) a learnable param, too):

$(x \diamond y)^{\diamond (i \phi z)} = x \diamond \big( (y \diamond (x \diamond z)) \diamond z \big)$

EDIT 2:

Another variant could be with 168 (look at the symmetry in the operation):

$x^{\diamond(\phi i y \diamond z)} = (y \diamond x) \diamond (x \diamond z)$

EDIT 3:

With 1076. 1076 is quite magical.

$x^{\diamond(\phi i y \diamond z)} = y \diamond ((x \diamond (x \diamond y)) \diamond y)$

[Alex Meiburg]

The notation is a bit mysterious, and the right hand side looks a bit arbitrary. Can you explain, maybe, how you came to this question? What prompted this?

[Alex Meiburg]

Oh, I see that the right hand side is 1043.

[Michael Bucko]

Nothing prompted this. It's an idea. I think such ideas still have to come from people. (That might change in some months or years).

The idea here is that the operation can be used as an exponent and transform $(x \diamond y)$ into 1043.

I posted about 1043 in 854 - it looks quite special. The reason for 1043 being quite special is this

2,4 -> 1043 -> 433 -> 854
2,4 -> 1043 -> 854

Also, in a way x's are on the left, z's are on the right, and y is "rather" in the middle, i.e. that's why I picked the LHS like this. It feels like unpacking-packing things, i.e. moving one thing in the bag a bit to the side, then moving the other one, and so on.

The reasoning that led to this is https://buildermath.substack.com/p/flower-algorithm -- it's about being able to efficiently grow a structured, long-context insight.

[Michael Bucko]

Btw. I create a lot of experiments like this (been doing that for a while), but they mostly land in the drawer.

One example you can find in my blog: https://buildermath.substack.com/p/on-fixed-width-rainbows (in this example, I know it's an not an ideal fixed width case (not a fixed width case at all), but I am still using that concept to explore the parameter space, and then eventually drop it to derive perhaps an unexpected conclusion).

[Michael Bucko]

Another variant of this could be:

$(x \diamond y)^{\diamond (iz)} = x \diamond \big( (y \diamond (x \diamond z)) \diamond z \big)$

[Michael Bucko]

Another variant of this could be:

$(x \diamond y)^{\diamond (i \phi z)} = x \diamond \big( (y \diamond (x \diamond z)) \diamond z \big)$