18 Equation 1729
In this chapter we study magmas that satisfy equation 1729,
for all \(x,y\). Using the squaring operator \(Sy := y \diamond y\) and the left and right multiplication operators \(L_y x := y \diamond x\) and \(R_y x = x \diamond y\), this law can be written as
This implies that \(L_y\) is injective and \(L_{Sy}\) is surjective, hence \(L_{Sy}\) is invertible. If \(y\) is a square (i.e., \(y \in SM\)), then \(L_y\) and \(L_{Sy}\) are both invertible, hence now \(R_y\) is also invertible, with inverse \(R_y^{-1} = L_y L_{Sy}\). We rewrite this as
for all \(y \in SM\).
We have the following procedure for extending a small magma \(SM\) satisfying Equation 1 to a larger one \(M\):
Let \(SM\) be a magma satisfying 1729, and let \(N\) be another set disjoint from \(SM\), and set \(M := SM \uplus N\). Suppose that we have a squaring map \(S': N \to SM\) (which will complement the existing squaring map \(S: SM \to SM\)), and bijections \(L'_a, R'_a: N \to N\) for all \(a \in SM\) (which will complement the existing bijections \(L_a, R_a: SM \to SM\) coming from \(SM\)), satisfying the following axioms:
(i) For all \(a \in SM\), we have \(L'_a = (R'_a)^{-1} (L'_{Sa})^{-1}\).
(ii) For all \(y \in N\), the elements \(R'_a y \in N\) are distinct from each other and from \(y\) as \(a \in SM\) varies.
(iii) If \(R'_a x = y\) for some \(a \in SM\) and some \(x,y \in N\), then \(L'_{S'y} L_{L_{S'x}^{-1} a} y = x\).
(iv) For all \(x \in N\), we have \((L'_{S'x})^2 x = x\).
Suppose also that we have an operation \(\diamond ': N \times N \to M\) satisfying the following axioms:
(v) For all \(x \in N\), we have \(x \diamond ' x = S'x\).
(vi) For all \(y \in N\) and \(a \in SM\), we have \(R'_a y \diamond ' y = L_{S'y}^{-1} a\).
(vii) For all \(x,y \in N\) with \(x \diamond ' y\) not already covered by rules (v) or (vi), we have \(x \diamond ' y = z\) for some \(z \in N\). Furthermore, \(z \diamond ' x = (L'_{S'x})^{-1} y\).
Then one can endow \(M\) with an operation \(\diamond '': M \times M \to M\) satisfying 1729 defined as follows:
If \(a,b \in SM\), then \(a \diamond '' b = a \diamond b\).
If \(a \in SM\) and \(x \in N\), then \(a \diamond '' b := L'_a b\).
If \(x \in N\) and \(a \in SM\), then \(b \diamond '' a := R'_a b\).
If \(x,y \in N\), then \(x \diamond '' y := x \diamond ' y\).
Furthermore, the 817 law \(x \diamond '' SS' x = x\) fails for any \(x \in N\).
We need to show that \(\diamond ''\) verifies the law Equation 1. In the case when \(x,y \in SM\), then the claim follows from the fact that \(SM\) already satisfied this equation. If \(x\) was equal to an element \(a \in SM\) and \(y \in N\), then by construction the law is equivalent to \(L'_{Sa} R'_a L'_a y = y\), which follows from axiom (i).
Now suppose that \(x \in N\) and \(y\) is equal to some element \(a\) of \(SM\). From axiom (v) we have \(x \diamond '' x = S'x\), and then this case of Equation 1 becomes
which follows from axiom (vi). So the only remaining case is when \(x,y \in N\). Using axiom (ii), we can divide into cases:
Case 1: \(x=y\). Then by (v) we need to show that \(L'_{S'x} L'_{S'x} x = x\), which follows from axiom (iv).
Case 2: \(y = R'_a x\) for some \(a \in SM\). Then by axiom (vi), we need to show that \(L'_{S'y} L'_{L_{S'x}^{-1} a} y = x\), which follows from axiom (iii).
Case 3: We are not in case 1 or case 2. Then by axiom (vii), we have \(y \diamond '' x = z\) for some \(z \in N\) with \(z \diamond '' y = (L'_{S'y})^{-1} x\). But this implies \(L'_{S'y} (z \diamond '' y) = x\), which is Equation 1.
We have now verified that \(\diamond ''\) satisfies 1729. For any \(x \in N\), we have \(x \diamond '' SS' x = R'_{SS' x} x\), and so the final claim follows from axiom (ii).
To build a magma satisfying 1729 but not 817, it thus suffices to produce
a 1729 magma \(SM\);
a set \(N\) of “non-squares”;
a squaring map \(S': N \to SM\);
bijections \(L'_a, R'_a: N \to N\) for all \(a \in SM\) satisfying the axioms (i)-(iv); and
an operation \(\diamond ': N \times N \to SM \uplus N\) satisfying the axioms (v)-(vii).
The magma \(SM\) is defined as follows:
Take \(SM\) to be a countably infinite abelian group of exponent \(4\), generated by generators \(E_n\) for \(n \in \mathbb {N}\) subject to the relations \(4E_n=0\).
\(SM\) is a 1729 magma, the squaring operation \(S: SM \to SM\) is just the doubling map \(Sa = 2a\), and the double squaring map \(S^2: SM \to SM\) is constant: \(S^2 a = 0\) for all \(a \in SM\).
Routine verification.
We now define \(N\), as well as some Cayley graph structures on it.
Take \(N\) to be the free non-abelian group with a generator \(e_a\) for each \(a \in SM\), thus \(N\) is the set of reduced words using the alphabet \(e_a\), \(e_a^{-1}\). Two elements \(x,y \in SM\) are said to be adjacent if \(x = e_a y\) or \(y = e_a x\) for some \(a \in SM\); this defines a left Cayley graph on \(N\). We make a partial ordering \(\leq \) on \(N\) by declaring \(y \leq x\) if \(y\) is a right subword of \(x\) (or equivalently, \(y\) is on the unique simple path from \(1\) to \(x\)). For instance, if \(a,b,c \in SM\) are distinct, then
If \(x \in N\) is not the identity, we define the parent of \(x\) to be the unique element \(y \in N\) adjacent to \(x\) whose reduced word is shorter. For instance, the parent of \(e_a e_b^{-1} e_c\) is \(e_b^{-1} e_c\).
\(N\) is countable, and \(\leq \) is a partial ordering.
Routine verification.
We will define the right multiplication operators \(R'_a: N \to N\) using the group action:
We set
for all \(a \in SM\) and \(x \in N\).
The operators \(R'_a\) are bijective and satisfy axiom (ii).
Routine verification.
We defer construction of the squaring map \(S': N \to SM\) for now, but turn to left-multiplication. From two applications of Equation 2 and the exponent 4 hypothesis we have
Thus, once \(L'_0\) is specified, we can define \(L'_a\) for all other \(a \in SM\) by the rule
Furthermore, from the \(a=0\) case of Equation 2 we must also have the axiom
(i’) \((L'_0)^2 = (R'_0)^{-1}\).
Conversely, we have
Suppose we have a bijection \(L'_0: N \to N\) that satisfies axiom (i’), and then define \(L'_a\) for all \(a \in SM\) by the formula 4. Then this recovers \(L'_0\) when \(a=0\) (to formalize this it may be convenient to give \(L'_0\) and \(L'_a\) distinct names), and the \(L'_a\) are all bijections and satisfy axiom (i). Furthermore, we have
for all \(a \in SM\).
Routine verification.
We now write the other remaining axioms in terms of \(L'_0\) rather than \(L'_a\) using Equation 4, Equation 5, Equation 3, and the magma law on \(SM\):
(iii’) If \(R'_a x = y\) for some \(a \in SM\) and some \(x,y \in N\), then \((R'_{S'y})^{-1} L'_0 R'_{2S'y} (R'_{a - S'x})^{-1} L'_0 R'_{2(a - S'x)} y = x\).
(iv’) For all \(x \in N\), we have \((R'_{S'x})^{-1} L'_0 R'_{2S'x} (R'_{S'x})^{-1} L'_0 R'_{2S'x} x = x\).
(v) For all \(x \in N\), we have \(x \diamond ' x = S'x\).
(vi’) For all \(y \in N\) and \(a \in SM\), we have \(e_a y \diamond ' y = a - S'y\).
(vii’) For all \(x,y \in N\) with \(x \diamond ' y\) not already covered by rules (v) or (v’), we have \(x \diamond ' y = z\) for some \(z \in N\). Furthermore, \(z \diamond ' x = (R'_{2S'x})^{-1} L'_0 e_0 e_{S'x} y\).
Suppose we can find a function \(S': N \to SM\), a bijection \(L'_0: N \to N\), and an operation \(\diamond ': N \times N \to SM \uplus N\) satisfying axioms (i’), (iii’), (iv’), (v), (vi’), (vii’). Then there exists a magma satisfying 1729 but not 817.
Construct the \(L'_a\) using Lemma 18.8. By Lemma 18.7 and direct verification we can noe verify axioms (i)-(vii), and then the claim follows from Theorem 18.1.
Our task is now to find a function \(S': N \to SM\), a bijection \(L'_0: N \to N\), and an operation \(\diamond ': N \times N \to M\) satisfying axioms (i’), (iii’), (iv’), (v), (vi’), (vii’).
We will again use a greedy construction for this, but with some modifications. Firstly, the axiom (i’), together with 3 means that we cannot restrict \(L'_0\) to be partially defined on just finitely many values: any relation of the form
for some \(x,y \in N\) would automatically imply that
and also
for all \(n \in \mathbb {Z}\). Thus, \(L'_0\) becomes defined on two right cosets \(\langle e_0 \rangle x\), \(\langle e_0 \rangle y\) of \(N\), where \(\langle e_0 \rangle := \{ e_0^n: n \in \mathbb {Z}\} \) is an infinite cyclic subgroup of \(N\). In general, we will require that \(L'_0\) is defined on a finite union of cosets of \(\langle e_0\rangle \).
In a somewhat similar vein, axiom (vii’), if iterated naively, would mean that a given entry \(x \diamond ' y = z\) of the multiplication table could potentially generate an infinite sequence of further entries, which unfortunately do not have as regular a pattern as the iterations Equation 6, Equation 7 of axiom (i’). So we will need to truncate this iteration by creating an addition category of “pending” identities \(I[x,y,z]\) of the form “\(z \diamond ' x = (R'_{2S'x})^{-1} L'_0 R'_0 e_{S'x} y\)” for some \(x,y,z \in N\), which will be temporarily undefined because \(S'x\) is undefined. More precisely,
A partial solution \((L'_0, \diamond ', S', {\mathcal I})\) is a collection of the following data:
A partially defined function \(L'_0: N \to N\), defined on a finite union of right cosets of \(\langle e_0\rangle \);
A partially defined operation \(\diamond ': N \times N \to M\), defined on a finite set;
A partially defined function \(S': N \to SM\), defined on a finite set; and
A finite collection \({\mathcal I}\) of “pending identities” \(I[x,y,z]\), which one can think of either as ordered triples of elements \(x,y,z \in N\), or as formal strings of the form “\(z \diamond ' x = (R'_{2S'x})^{-1} L'_0 R'_0 R'_{S'x} y\)” for some \(x,y,z \in N\).
Furthermore, the following axioms are satisfied:
(i”) \(L'_0 x\) is defined and equal to \(y\), then we have the identities Equation 6, Equation 7 for all \(n\in \mathbb {Z}\).
(S) If \(S'x\) is defined for some \(x \in N\), then \(S'y\) is defined for all \(y \leq x\).
(iii”) If \(R'_a x = y\) for some \(a \in SM\) and some \(x,y \in N\), and \(S'x, S'y\) are defined, then \((R'_{S'y})^{-1} L'_0 R'_{2S'y} (R'_{a-S'x})^{-1} L'_0 R'_{2(a-S'x)} y\) is defined and equal to \(x\).
(iv”) If \(x \in N\) is such that \(S'x\) is defined, then \((R'_{S'x})^{-1} L'_0 R'_{2S'x} (R'_{S'x})^{-1} L'_0 R'_{2S'x} x\) is defined and equal to \(x\).
(v”) If \(x \in N\) and \(x \diamond ' x\) is defined, then \(S'x\) is defined and equal to \(x \diamond ' x\).
(vi”) For all \(y \in N\) and \(a \in SM\), if \(R'_a y \diamond ' y\) is defined, then \(a - S'y\) is defined and equal to \(R'_a y \diamond ' y\).
(vii”) For all \(x,y \in N\) and \(x\) is not equal to \(y\) or \(R'_a y\) for any \(a \in SM\), and \(x \diamond ' y\) is defined, then it is equal to some \(z \in N\). Furthermore, either \(I[x,y,z]\) is a pending identity, or else \(z \diamond ' x\) and \((R'_{2S'x})^{-1} L'_0 R'_0 R'_{S'x} y\) are defined and equal to each other.
(P) If \(I[x,y,z]\) is a pending identity, then \(x,y,z \in N\), and \(Sx\) and \(z \diamond ' x\) are undefined. Furthermore, \(z\) is not equal to \(x\) or \(R'_a x\) for any \(a \in SM\), and \(y\) is not of the form \((R'_0)^n x\) or \((R'_0)^n y_0\) for any \(n\), where \(y_0\) is the parent of \(x\).
(P’) If \(I[x,y,z]\) and \(I[x,y',z]\) are pending identities, then \(y=y'\).
(P”) If \(I[x,y,z]\) is a pending identity, then \(x \diamond ' y = z\).
(L) If \(y\) is the parent of \(x\) with \(x = R'_a y\), and \((R'_{a-S'y})^{-1} L'_0 R'_{2(a-S'y)} x\) is defined, then it is not equal to \(x\).
We say that one partial solution \((\tilde L'_0, \tilde\diamond ', \tilde S', \tilde{\mathcal I})\) extends another if \((L'_0, \diamond ', S', {\mathcal I})\) if \(\tilde L'\) is an extension of \(L'_0\), \(\tilde\diamond '\) is an extension of \(\diamond '\), and \(\tilde S'\) is an extension of \(S'\). (No constraint is imposed on the final components \(\tilde{\mathcal I}, {\mathcal I}\).) This is a preordering.
There exists a partial solution.
Set \(L'_0, \diamond ', S'\) to be empty functions, and have the set of pending identities to also be empty. The verification of the required axioms is then routine.
Suppose that one has a sequence \((L'_{0,n}, \diamond '_{n}, S'_{n}, {\mathcal I}_n)\) of partial solutions, each one an extension of the previous, such that for any \(x, y \in N\), \(L'_{0,n} x\), \(x \diamond '_{n} y\), and \(S'_n x\) are defined for some \(n\). Then there exists a 1729 magma that does not satisfy 817.
Take the direct limit of the chain to obtain total functions \(L'_0, \diamond ', S'\). The axioms (i’), (iii’), (iv”), (v”), (vi”), (vii”) of the partial solutions then easily imply that the direct limit satisfies the axioms (i’), (iii’), (iv’), (v), (vi’), (vii’) (one also uses axiom (P) to note that all pending identities disappear in the direct limit). The claim now follows from Lemma 18.9.
Now we seek to enlarge a partial solution. We first make an easy observation:
Suppose one has a partial solution in which \(L'_0 x\) is undefined for some \(x \in N\). Then one can extend the partial solution so that \(L'_0 x\) is now defined.
By axiom (i”), \(L'_0 (R'_0)^n x\) is undefined for every integer \(n\). Let \(d = E_m\) be a generator of \(SM\) that does not appear as a component of any index of any of the generators \(e_a\) appearing anywhere in the partial solution; such a \(d\) exists due to the finiteness hypotheses. We set \(L'_0 x := e_d\), and then extend by Equation 6, Equation 7, thus
and
Because of the new nature of \(d\), no collisions in the partial function \(L_0\) are created by this operation. It is then easy to check that axiom (i”) is preserved by this operation, whereas none of the other axioms (S), (iii”), (iv”), (v”), (vi”), (vii”), (P), (P’), (P”) are affected by this extension. With some effort, axiom (L) can also be verified.
As a corollary, we have
Suppose one has a partial solution. Let \(A\) be a finite subset of \(N\). Then one can extend the partial solution so that \(L'_0 x\) is now defined for all \(x \in A\).
Iterate Proposition 18.13 in the obvious fashion.
Next, we provide a tool for enlarging the domain of definition of \(S'\). The main step is the following inductive one with extra axioms.
Suppose one has a partial solution in which \(S'x\) is undefined for some \(x \in N\), but \(S'y\) is defined for all \(y {\lt} x\). (This hypothesis is vacuous for \(x=1\).) Let \(y_0\) be the parent of \(x\) (if \(x \neq 1\)), and assume the following additional axioms:
(A) If \(R'_a x = y_0\) for some \(a \in SM\), then \(L'_0 R'_{S' y_0} x\) is defined.
(B) If \(x = R'_a y_0\) for some \(a \in SM\), then \(L'_0 R'_{2(a-S'y_0)} x\) is defined.
(C) If \(I[x,y,z]\) for some \(y,z \in N\), and \(S'z\) is defined, then \(L'_0 R'_0 R'_{S'z} x\) is defined.
Then one can extend the partial solution so that \(S'x\) is now defined.
Let \(d_0, d_1 \in SM\) be generators \(E_{n_0}\) of \(SM\) that do not appear in the index \(a\) of any \(e_a\) that currently appears in the partial solution (of which there are only finitely many). We also set some further distinct generators \(d'_{y,z} \in SM\) for \(y,z \in SM\) that are distinct from each other and all previous generators (this is possible as we have infinitely many generators). We set \(S'x, x \diamond ' x := d_0\); this preserves axiom (S) and (v”). We set
and then extend these choices using Equation 6, Equation 7 to recover axiom (i’), thus for instance
and
for all \(n \in \mathbb {Z}\).
To retain axiom (iii”), we perform the following actions:
If \(y_0\) is undefined, do nothing.
If \(R'_a x = y_0\) for some \(a \in SM\) (so that \(L'_0 R'_{S'y_0} x\) is defined, by hypothesis (A)), set \(L'_0 R'_{2(a-d_0)} y_0 := R'_{a-d_0} (R'_{2S'y_0})^{-1} R'_0 L'_0 R'_{S'y_0} x\). Then extend using Equation 6, Equation 7.
If \(x = R'_a y_0\) for some \(a \in SM\) (so that \(L'_0 R'_{2(a-Sy_0)} x\) is already defined, by hypothesis (B)), set \(L'_0 R'_{2d_0} (R'_{a-S'y_0})^{-1} L'_0 R'_{2(a-Sy_0)} x :=R'_{d_0} y_0\), and extend using Equation 6, Equation 7.
To retain axiom (P), we perform the following actions for each pending identity of the form \(I[x,y,z]\) for some \(y,z\), executed in some arbitrary order.
Remove this identity \(I[x,y,z]\) from the list of pending identities.
Set \(L'_0 R'_0 R'_{d_0} y := e_{d'_{y,z}}\), and \(x' \diamond y' := z'\), where \((x',y',z') := (z, x, (R'_{2d_0})^{-1} e_{d'_{y,z}})\). Extend all the new definitions of \(L'_0\) using Equation 6, Equation 7.
If \(S'x'\) is undefined, add \(I[x',y',z']\) as a pending identity.
If instead \(S'x'\) is defined (which makes \(L'_0 R'_0 R'_{S'x'} y'\) defined, by axiom (C)), set \(x'' \diamond y'' = z''\), where \((x'',y'',z'') := (z', x', (R'_{2S'x'})^{-1} L'_0 R'_0 R'_{S'x'} y')\). Then add \(I[x'',y'',z'']\) as a pending identity.
One can check that these definitions do not cause any collisions in the partial function \(L'_0\), and that axioms (i’), (iii’), (iv”), (v”), (vii”), (P), (P’), (P”) are preserved; the remaining axiom (vi”) is unaffected by this extension. Axiom (L) can also be shown to be preserved after a tedious calculation.
Suppose one has a partial solution in which \(S'x\) is undefined for some \(x \in N\), but \(S'y\) is defined for all \(y {\lt} x\). (This hypothesis is vacuous for \(x=1\).) Then one can extend the partial solution so that \(S'x\) is now defined.
Let \(y_0\) be the parent of \(x\), that is to say the unique neighbor of \(x\) in the path to \(1\) (this is only defined for \(x \neq 1\)), then by axiom (i”) \(y_0\) is the unique neighbor for which \(S'y_0\) is defined, and we either have \(x = R'_a y_0\) or \(R'_a x = y_0\) for some unique \(a \in SM\).
By using Proposition 18.14, we may impose axioms (A), (B), (C) without loss of generality, as this only imposes a finite set of conditions. The claim now follows from Proposition 18.15.
As a corollary, we have
Suppose one has a partial solution in which \(S'x\) is undefined for some \(x \in N\). Then one can extend the partial solution so that \(S'x\) is now defined.
Obtained by induction from Proposition 18.16, using the fact that there are no infinite descending chains in \(N\).
Finally, we give a tool for enlarging \(\diamond \):
Suppose one has a partial solution in which \(x \diamond ' y\) is undefined for some \(x,y \in N\). Then one can extend the partial solution so that \(x \diamond ' y\) is now defined.
By applying Proposition 18.17 as needed, we may assume without loss of generality that \(S'x\) and \(S'y\) are already defined (among other things, this removes the possibility that \(x \diamond ' y\) is part of a pending identity). If this makes \(x \diamond ' y\) defined, then we are done, so we may assume that this is not the case.
Similarly, by using Proposition 18.13, we may assume without loss of generality that \(L'_0 R'_0 R'_{S'x} y\) is defined.
We now divide into cases:
Case 1: \(x=y\). In this case we set \(x \diamond ' y := S'x\). It is clear that this preserves axiom (v”), and no other axiom is impacted.
Case 2: If \(x = R'_a y\) for some \(a \in SM\), then we set \(x \diamond ' y := a - S'y\). This preserves axiom (vi”), and no other axiom is impacted.
Case 3: If \(x\) is not equal to \(y\) or \(R'_a y\) for any \(a \in SM\). Let \(d_0 \in SM\) be a generator that does not appear as a component of any index of any of the generators \(e_a\) appearing anywhere in the partial solution. We set \(x \diamond ' y := z\) with \(z := e^2_{d_0}\).
This temporarily disrupts axiom (vii”). To recover it, we perform the following actions.
Set \(x' \diamond ' y' = z'\), where \((x',y',z') := (z, x, (R'_{2S'x})^{-1} L_0 R'_0 R'_{Sx} y)\).
Add \(I[x',y',z']\) as a pending identity.
One can check that these definitions do not cause any collisions in the partial function \(L'_0\), and that axioms (i’), (vii’), (P), (P’) are preserved; the remaining axioms (S), (iii”), (iv”), (v”), (vi”) are unaffected by this extension.
There exists a magma that satisfies equation 1729 but not equation 817.
Starting from Lemma 18.11 and applying Proposition 18.13, Proposition 18.17, Proposition 18.18 alternatingly, one can find a chain of partial solution that are total in the limit. The claim now follows from Lemma 18.12.