Which algebraic structure is like a magma augmented with an operation which is an anti-function?

Question

Is there any structure similar to a free magma, including an operation which splits, or undoes, the binary operation?

A set S together with two operations (S, •, ~)

• : S × S → S
~ : S → S × S

Where ⋅(∼(x)) = x

Background: If free magmas abstractly describe binary tree structures, they don't provide a mechanism to opaquely decompose nodes for examination. Which structure does?

Answer 1

You're looking for the Jónsson-Tarski algebras. Given a set $X$ together with a bijection $m:X\times X\to X$, we have the reverse map $X\to X\times X$ and the left and right components can be given by $\ell:X\to X$ and $r:X\to X$. In other words it satisfies the identities:

$m(r(x),\ell(x))=x$, $r(m(x,y))=x$, and $\ell(m(x,y))=y$.

Some interesting facts about Jónsson-Tarski algebras include:

1. The only finite Jónsson-Tarski algebra is trivial,
2. the free Jónsson-Tarski algebra on $n$ generators is isomorphic to the free Jónsson-Tarski algebra on $m$ generators, and
3. the automorphism group of the finitely generated free Jónsson-Tarski algebra is Thompson's group $V$.