We may assume that \(E\) does not already contain any pair of the form \((a_0,c)\), since we are done otherwise. By Axiom 1 implies that \(a_0 \neq 0\). Let \(b_1,\dots ,b_n\) denote all the integers \(b_i\) such that \((b_i,a) \in E\), then \(n \geq 0\) is finite, and by Axiom 2 all the \(b_i\) are non-zero and not equal to \(-a\). Let \(c_0\) be a sufficiently large integer to be chosen later. We then add the pairs \((a_0,c_0)\) and \((c-b_i,-b_i)\) to \(E\) for all \(i\). Furthermore, if \(i\) is such that \((-b_i,d_i) \in E\) for some (necessarily unique and non-zero) \(d_i\), we also add \((d_i+b+i-c_0, b_i-c_0)\) to \(E\). Let \(E'\) be the resulting set of pairs. Axioms 4, 5 for \(E\) ensure that (for \(c_0\) large enough) the addition of these pairs do not cause a violation of Axioms 2 or 3 for \(E'\), and of course Axiom 1 for \(E'\) will also be retained. As for Axiom 4, one can check that the only new pairs of pairs \((',b), (b,c)\) that would trigger these axioms either take the form \((b_i,a_0), (a_0,c_0)\) or \((c_0-b_i,-b_i), (-b_i,d_i)\), and in either case we see from construction that Axiom 4 remains in effect for \(E'\). Finally, the new quadruples of pairs \((b,a), (b',a), (-b, d), (-b',d') \in E'\) only arise when either \((-b,d)\), \((-b',d')\) is equal to \((a_0,c_0)\) and the other three pairs in the quadruple were already in \(E\), and for \(c_0\) large enough we see that Axiom 5 remains valid.

If we choose \(a\) sufficiently large, and set \(a' = 2a\) and \(a''=3a\) (say), the claim simply follows by adding \((a,a+h)\), \((a',a'+h)\) and \((a'',a''+h)\) to \(E\) and verifying that none of the axioms are violated.

By Lemma 17.2, Lemma 17.3 and the greedy algorithm starting with the seed consisting of \((0,0)\), \((-1,2)\), \((3,1)\), we obtain a graph \(\{ (a,f(a)): a \in \mathbb {Z}\} \) of a function \(f:\mathbb {Z}\to \mathbb {Z}\) with \(f(0)=0\), \(f(-1)=2\), \(f(3)=1\) and the property that if \(f(a)=b\) and \(f(b)=c\), then \(f(c-a)=-a\), and also with the property that for every non-zero \(h\) there are distinct \(a_h, a'_h, a''_h\) with \(f(a_h)=a_h+h\), \(f(a'_h) = a'_h+h\), and \(f(a''_h) = a''_h+h\). We then have Equation 3 holds. If we then define the magma operation \(\diamond \) by Equation 2, we obtain Equation 1. Also, from construction we see that \(R_d (d - a_h) = R_d (d - a'_h) = R_d (d - a''_h) = d+h\) for any \(d,h\), giving the second claim. Finally, from construction we have \(R_a (a+1) = a+1+f(-1) = a+3\), so \(L_a R_a (a+1) = a + f(3) = a+1\), giving the final claim.

Let \(y = (a,c,n)\) be a “non-square” in \(G'\), thus \(a \neq c\) and \(Sy = a\). We first define the operation \(L_a\) on \(y\) as follows:

• If \(n \neq 0\), we set \(L_a y := (a,c,0)\).

• If \(n=0\), we set \(L_a y = a\).

In particular, we obtain the special case \(L_{Sy} L_a L_a y = a\) of Axiom B. We also see that the the preimage \(\{ z \in G': L_a z = y \} \) is already infinite if \(n = 0\), but this has not yet been established in the \(n \neq 0\) case.

We still need to define \(L_c y\) for other \(c \in \mathbb {Z}\). We claim that for each \(b \in \mathbb {Z}\) we can find \(c_{y,b} \in \mathbb {Z}\), which are distinct as \(b\) varies, such that \(L_a L_{c_{y,b}} b = c_{y,b}\). For \(b=a\), we can write \(L_a L_{c_{y,b}} b = L_a R_a c_{y,b}\) and invoke the final property of Corollary 17.4. For \(b \neq a\), we apply Corollary 17.4 to find \(c_{y,b} \in \mathbb {Z}\) distinct from \(b, c_{y,a}\) such that \(R_a c_{y,b} = b\); note that \(c_{y,b}\) is also distinct from \(a\) since \(R_a a = a \neq b\). Clearly the \(c_{y,b}\) are distinct as \(b\) varies. We now have \(L_a L_{c_{y,b}} b = L_{Sa} L_{c_{y,b}} L_{c_{y,b}} a = c_{y,b}\) as desired.

So if we define \(L_{c_{y,b}} y := b\) for all \(b \in \mathbb {Z}\), then we have \(L_{Sy} L_{c'} L_{c'} y = c'\) whenever \(c'\) is of the form \(c' = c_{y,b}\) for some \(b \in \mathbb {Z}\). But we still have \(L_{c'} y\) undefined if \(c'\) is not of this form or equal to \(a\), in particular, \(L_c y\) is currently not defined.

We also note that with our construction so far, that \(L_c y \neq c\) whenever \(y \in G'\) and \(c \in \mathbb {Z}\) is distinct from \(Sy\). This property will be preserved in the construction below (because all future assignments of \(L_c y\) will take values in \(G'\) rather than \(\mathbb {Z}\)).

Now we greedily fill in the rest of the \(L_{c'}\) functions. Suppose at some stage of this process, we have a non-square element \(y = (a,c,n)\) for which \(L_{c'} y\) is not yet defined for some \(c'\), then \(c' \neq a\). We can use the surjectivity of \(L_a: \mathbb {Z}\to \mathbb {Z}\) to write \(c' = L_a b\) for some \(b \in \mathbb {Z}\), then set \(L_{c'} y = b\); then we have \(L_{Sy} L_{c'} L_{c'} y = c'\).

Next, suppose we want to write \(y\) as \(L_{c'} z\) for some \(z\) where \(L_{c'}\) is not currently defined, in such a way that \(L_{Sz} L_{c'} L_{c'} z = c'\). There are several cases:

• Case 1: \(L_{c'} y = w\) for some \(w \in G'\). By construction, \(c_{w,c'}\) is distinct from \(c'\) and \(L_{c_{w,c'}} w = c'\). Set \(z = (c_{w,c'}, c', n', 0)\) where \(n'\) is large enough that \(L_{c'} z\) has not yet been assigned. Then set \(L_{c'} z = y\), so that we have \(L_{Sz} L_{c'} L_{c'} z = c'\) as required.

• Case 2: \(c' \neq Sy\) and \(L_{c'} y = b\) for some \(b \in \mathbb {Z}\). By the previous property, \(c' \neq b\). By Corollary 17.4, we can find \(a'\) such that \(R_b a' = c'\) and \(a' \neq c'\). Then set \(z = (a', c', n', 0)\) where \(n'\) is large enough that \(L_{c'} z\) has not yet been assigned. Then set \(L_{c'} z = y\), so that we have \(L_{Sz} L_{c'} L_{c'} z = c'\) as required.

• Case 3: \(c' \neq Sy\) and \(L_{c'} y\) is currently undefined. Use the previous procedure to set \(L_{c'} y\) equal to some \(b \in \mathbb {Z}\), then apply the Case 2 analysis.

• Case 4: \(c' = Sy\). We can assume \(n=0\), since otherwise we are in Case 1 by construction. But then we do not need to locate any \(z\) because, as previously mentioned, the set \(\{ z \in G': L_{c'} z = y \} \) is already infinite.

Iterating these procedures over a well-ordering of \(G' \times \mathbb {Z}\times \mathbb {N}\), we can show that for any finite initial segment in this well-ordering, we can create a partially defined extension of all the \(L_b\), such that if \((x,c',m)\) is in this initial segment, then \(L_{Sx} L_{c'} L_{c'} x\) is well-defined and equal to \(x\), and \(\{ y \in G': L_b y = x \} \) has at least \(m\) elements. Running this iteration over the entire well-ordering, we obtain the claim.

We can work with a single \(x \in G'\). Our task is to find a function \(L_x\) for which

and

for all \(y \in G\) (note that \(L_{Sy}\) is already fully constructed).

Define a seed to be an injective partial function \(L_x\) defined on finitely many values and obeying Equation 4, as well as Equation 5 whenever \(L_x L_x y\) is defined. If there is a \(y \in G\) for which \(L_x y\) is currently undefined, then by hypothesis \(y\) is equal to \(L_x z\) for at most one \(z\). If such a \(z\) exists, we set \(L_x y\) to be an element of \(\{ w: L_{Sz} w = x \} \) that is not already in the domain or range of \(L_x\); if no such \(z\) exists, we set \(L_x y\) to be arbitrary element of \(G\) not already in the domain or range. In either case, we see that the seed property is preserved. Starting from the seed in which \(L_x\) is only defined on \(x\) and maps it to \(Sx\), we obtain the claim.

We perform the above construction, but with one refinement. Pick some \(x \in G'\). Then after the constructions in Proposition 17.5, \(L_{Sx} x\) is defined and equal to some element \(y\) distinct from \(x\). If \(y \in \mathbb {Z}\) then \(L_y x \neq x\) by construction and we are done, so suppose \(y \in G'\). It is then a routine matter to modify the construction in Proposition 17.6 to ensure that \(L_y x \neq x\), by adding one more element to the seed for \(L_y\) to set \(L_y x\) to some arbitrary value not equal to \(x, y, Sy\).