A Funny Definition of Groups

I was reading this paper of Oliver Lorscheid, " Towards the horizons of Tits's vision -- on band schemes, crowds and F1-structures" on his recent work doing algebraic geometry over $\mathbb{F}_1$, and I rabbitholed a bit on his definition of "crowd", a generalization of groups.

The basic idea is we relax the group operation to be more relational than functional. Where a group says "given $a$ and $b$ we have a product $ab$" we instead say "given $a, b, c$, perhaps they have the property that we would otherwise describe as $abc = 1$, if we had a group product". So a crowd is a set $G$ with a distinguished element 1 and a ternary relation $R \subseteq G^3$ satisfying some axioms \[x = 1 \iff (x, 1, 1)\] \[(x, y, 1) \iff (y, x, 1)\] \[(x, y, z) \iff (z, x, y)\] So this is kind of like a group, but the multiplication can be multi-valued. Then you get to prove the nice theorem that if you additionally impose a single-valuedness axiom like \[ \forall x, y. \exists ! z . (x, y, z)\] you have an actual honest-to-goodness group... or at least that's what I assumed, until I noticed Lorscheid's Proposition 5.7, whose condition (2) is basically imposing associativity. I found this a bit disappointing, as I hoped associativity should somehow come from the existing axioms for free. I guess it doesn't.

But! It was therefore a nice open-ended puzzle to come up with a variation on the definition of crowd that did somehow imply associativity if you tried to define groups in terms of it.

Multicrowds

Here's my attempt at doing exactly that. I think it helps me see why I was overoptimistic to expect associativity to be latent in Lorscheid's definition. Say a multicrowd is given by a set $G$, together with, for each $n \in \N$, a relation $R_n \subseteq G^n$, written $(a_1, \ldots, a_n)$ for $a_1, \ldots, a_n \in G$. There are three axioms: \[ {(a_1, \ldots, a_n) \over (a_n, a_1, \ldots, a_{n-1})}\pi\] \[ { (a_1, \ldots, a_n) \qquad (b_1, \ldots, b_m) \over (a_1, \ldots, a_n, b_1, \ldots, b_m)}\iota\] \[ { (a_1, \ldots, a_n, b_1, \ldots, b_m) \qquad (b_1, \ldots, b_m) \over (a_1, \ldots, a_n)}\delta\] The axiom $\pi$ says we can cyclically permute facts in the relation, $\iota$ says we can insert a related tuple, and $\delta$ says we can delete a related tuple. Note that insertions and deletions seem to be done only at the end of tuples, but permutation can be used to insert them anywhere.

Say a multicrowd is functional if for every $n$ and $a_1, \ldots, a_n \in G$, there is a unique $b$ such that $(a_1, \ldots, a_n, b)$. Then we should have

Theorem: A functional multicrowd is the same thing as a group.
Proof: To make a functional multicrowd from a group, define \[(a_1, \ldots, a_n) \iff a_1\cdots a_n = 1\] and check $\pi, \iota, \delta$ and functionality are satisfied.

In the other direction, to make a group from a functional multicrowd, we first define some notation. Let \[[a_1, \ldots, a_n]\] be the unique $b$ such that $(a_1, \ldots, a_n)$, and define the group structure $(G, \cdot, 1, {—}^{-1})$ by \[1 = [] \qquad a^{-1} = [a] \qquad a\cdot b = [[a,b]]\] Most of the group axioms are easy to show. Let's show associativity. We must show $(ab)c = a(bc)$, in other words \[[[[[a,b]],c]] = [[a, [[b,c]]]] \tag{*}\] First of all notice that \[ (a, [[b, c]], [a,[[b,c]]])\] holds by definition of $[\cdots]$. Since $(b, c, [b,c])$, we can insert it to obtain \[(a, b, c, [b, c], [[b, c]], [a,[[b,c]]])\] Since $[b, c], [[b, c]]$, we can delete it to obtain \[(a, b, c, [a,[[b,c]]])\] By the uniqueness part of functionality, we have shown \[[a, b, c] = [a,[[b,c]]]\] So it would suffice to show that also \[[a, b, c] = [[[a,b]],c] \tag{**}\] We can do this as follows. First take \[ ( [[a,b]], c, [[[a,b]],c])\] which is true by definition. Insert $(a, b, [a, b])$ to get \[ (a, b, [a, b], [[a,b]], c, [[[a,b]],c])\] Now delete $([a, b], [[a,b]])$ to get \[ (a, b, c, [[[a,b]],c])\] Hence we have shown $(**)$ and it follows that $(*)$, as required. $\cqed$

Why did this definition give us associativity?

It was critical that we had more "room" in our tuples to move things around, and also my axioms intrinsically involve list concatenation, which is another way of sneaking associativity in. But I still find these axioms pleasant and symmetric.