It suffices to prove that Equation 1 implies Equation 2. Write \(w = y \diamond z\), then from Equation 1 we can write \(z = z_1 \diamond z_2\) with \(z_1 = z \diamond z\) and \(z_2 = z \diamond (z \diamond z)\), and then by Equation 1

From another application of Equation 1 we have

as required.

If \(x \to y\) then \(y = x \diamond z\), then writing \(z = z_1 \diamond z_2\) as before we obtain

giving the forward implication. The backwards implication follows by duality.

If \(a \to b \to c\) is good then \(b = a \diamond c\); if \(c \to d \to e\) is good then \(d = c \diamond e\); and if \(e \to a\) then \(a = e \diamond z\) for some \(z\) by definition. By Equation 2 we then have

so \(b \to c \to d\) is good.

Define an operation \(\diamond : G \times G \to G\) by defining \(x \diamond y\) to be the unique vertex \(z\) for which one has a good path \(x \to z \to y\); this is well-defined by Claim 1, and by Claims 2-3, the property \(x \to y\) holds if and only if \(y = x \diamond z\) for some \(z\), and also if and only if \(x = w \diamond y\) for some \(w\). In particular, for all \(x,y,z\), we have a \(5\)-cycle

with \(y \to y \diamond x \to x\) and \(x \to x \diamond (z \diamond y) \to (z \diamond y)\) good, hence by Claim 4 we have Equation 1 as required.

By the previous comments, we can ignore Claim 4 as it is automatic, and focus on completing the partial weak central groupoid on \(G\) to a complete weak central groupoid by ensuring Claims 1, 2, 3 hold. By the usual greedy algorithm, it suffices to show that any individual failure of Claim 1, 2 or 3 can be resolved by adding some finite number of edges to the graph.

Suppose first that Claim 2 fails, that is to say the partial weak central groupoid contains an edge \((a,n) \to (b,m)\) that is not the initial segment of any good path. Since the base relaxed weak central groupoid \(G_0\) obeys Claim 2, we can find a good path \(a \to b \to c\) in the base. We then pick a natural number \(l\) not previously occurring in the partial weak central groupoid, and adjoin the edge \((b,m) \to (c,l)\) to that partial weak central groupoid. All new paths created in this way are declared good or bad depending on whether their images in \(G_0\) are good or bad, in particular \((a,n) \to (b,m) \to (c,l)\) is good. This can be checked to still be a partial extension of \(G_0\) (no violation of Claim 1” is created), and now Claim 2 is resolved at for the edge \((a,n) \to (b,m)\). A similar argument permits one to resolve any violations of Claim 3.

If Claim 1 is violated, then there is a pair \((a,n), (b,m)\) that currently has no good path of length two in the partial weak central groupoid. As the base relaxed weak central groupoid \(G_0\) obeys Claim 1’, we can find a good path \(a \to c \to b\) in \(G_0\). We then pick a natural number \(l\) not previously occurring, and adjoin the edges \((a,n) \to (c,l) \to (b,m)\). All new paths created in this way are declared good or bad depending on whether their images in \(G_0\) are good or bad, in particular \((a,n) \to (c,l) \to (b,m) \to (c,l)\). One can check that this is still a partial extension of \(G_0\) (no violation of Claim 1” is created), and now Claim 1 is resolved at the pair \((a,n), (b,m)\).

Computer check reveals that the carrier \(G_0=\{ 0,1,2,3 ,4\} \) with incidence matrix

is a relaxed weak central groupoid if we declare the paths \(0 \to 0 \to 0\), \(0 \to 0 \to 1\), \(0 \to 1 \to 1\), \(1 \to 0 \to 0\), \(1 \to 1 \to 0\), \(1 \to 1 \to 1\) to be bad, and all other paths in the directed graph to be good. We can also check the following axioms:

• Anti-3457: There exist \(x,y,z,w\) with \(x \to z \to x\), \(z \to w \to y\) both good, and \(x \to z \to w\) bad. (One can take \(x=1\), \(y=4\), \(z = 0\), \(w = 0\).)

• Anti-2087: There exist \(x,y,z,w,u\) with \(y \to z \to x\), \(z \to w \to x\), and \(x \to u \to x\) good, and \(w \to x \to u\) is bad. (One can take \(x=1, y=2\), \(z=4\), \(w=1\), \(u=0\).)

• Anti-2124: There exists \(x,y,z,w,u\) with \(y \to z \to y\), \(z \to w \to x\) and \(x \to u \to x\) good, and \(w \to x \to u\) bad. (One can take \(x=1\), \(y=2\), \(z=4\), \(w = 1\), \(u=0\).)

• Anti-3511: There exists \(x,y,z,w\) with \(x \to z \to y\) and \(z \to w \to x\) good, and \(x \to z \to w\) bad. (One can take \(x=1\), \(y=3\), \(z = 1\), \(w=0\).)

Let \(G_*\) be a finite partial extension of \(G_0\) to be chosen later. By Proposition 12.5, we can complete this to a complete weak central groupoid \(G\) with carrier \(G_0 \times \mathbb {N}\). Depending on how we choose \(G_*\), we can ensure that this \(G\) refutes one of the four laws 3457, 2087, 2124, 3511:

• Refuting 3457: Let \(x,y,z,w\) be as in the claim Anti-3457, then select \(G_*\) to be the directed graph with edges \((x,0) \to (z,2) \to (x,0) \to (z,2) \to (w,3) \to (y,1)\). One can check that this is a partial extension, and that \(G\) will refute 3457 with \(x,y\) replaced by \((x,0),(y,1)\).

• Refuting 2087: Let \(x,y,z,w,u\) be as in the claim Anti-2087, then select \(G_*\) to be the directed graph with edges \((y,1) \to (z,2) \to (x,0)\) and \((z,2) \to (w,3) \to (x,0) \to (u,4) \to (x,0)\). One can check that this is a partial extension, and that \(G\) will refute 2087 with \(x,y\) replaced by \((x,0),(y,1)\).

• Refuting 2124: Let \(x,y,z,w,u\) be as in the claim Anti-2124, then select \(G_*\) to be the directed graph with edges \((y,1) \to (z,2) \to (y,1)\) and \((z,2) \to (w,3) \to (x,0) \to (u,4) \to (x,0)\). One can check that this is a partial extension, and that \(G\) will refute 2124 with \(x,y\) replaced by \((x,0),(y,1)\).

• Refuting 3511: Let \(x,y,z,w\) be as in the claim Anti-3511, then select \(G_*\) to be the directed graph with edges \((x,0) \to (z,2) \to (y,1)\) and \((z,2) \to (w,3) \to (x,0)\). One can check that this is a partial extension, and that \(G\) will refute 3511 with \(x,y\) replaced by \((x,0),(y,1)\).