18 Equation 1729

In this chapter we study magmas that obey equation 1729,

x = (y ⋄ y) ⋄ ((y ⋄ x) ⋄ y) .

for all $x, y$ . Using the squaring operator $S y := y ⋄ y$ and the left and right multiplication operators $L_{y} x := y ⋄ x$ and $R_{y} x = x ⋄ y$ , this law can be written as

L_{S y} R_{y} L_{y} x = x .

This implies that $L_{y}$ is injective and $L_{S y}$ is surjective, hence $L_{S y}$ is invertible. If $y$ is a square (i.e., $y \in S M$ ), then $L_{y}$ and $L_{S y}$ are both invertible, hence now $R_{y}$ is also invertible, with inverse $R_{y}^{- 1} = L_{y} L_{S y}$ . We rewrite this as

L_{y} = R_{y}^{- 1} L_{S y}^{- 1}

for all $y \in S M$ .

We have the following procedure for extending a small magma $S M$ obeying Equation 1 to a larger one $M$ :

Theorem 18.1 Extending a 1729 magma

Let $S M$ be a magma obeying 1729, and let $N$ be another set disjoint from $S M$ , and set $M := S M ⊎ N$ . Suppose that we have a squaring map $S^{'} : N \to S M$ (which will complement the existing squaring map $S : S M \to S M$ ), and bijections $L_{a}^{'}, R_{a}^{'} : N \to N$ for all $a \in S M$ (which will complement the existing bijections $L_{a}, R_{a} : S M \to S M$ coming from $S M$ ), obeying the following axioms:

(i) For all $a \in S M$ , we have $L_{a}^{'} = (R_{a}^{'})^{- 1} (L_{S a}^{'})^{- 1}$ .
(ii) For all $y \in N$ , the elements $R_{a}^{'} y \in N$ are distinct from each other and from $y$ as $a \in S M$ varies.
(iii) If $R_{a}^{'} x = y$ for some $a \in S M$ 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 $⋄^{'} : N \times N \to M$ obeying the following axioms:

(v) For all $x \in N$ , we have $x ⋄^{'} x = S^{'} x$ .
(vi) For all $y \in N$ and $a \in S M$ , we have $R_{a}^{'} y ⋄^{'} y = L_{S^{'} y}^{- 1} a$ .
(vii) For all $x, y \in N$ with $x ⋄^{'} y$ not already covered by rules (v) or (vi), we have $x ⋄^{'} y = z$ for some $z \in N$ . Furthermore, $z ⋄^{'} x = (L_{S^{'} x}^{'})^{- 1} y$ .

Then one can endow $M$ with an operation $⋄^{″} : M \times M \to M$ obeying 1729 defined as follows:

If $a, b \in S M$ , then $a ⋄^{″} b = a ⋄ b$ .
If $a \in S M$ and $x \in N$ , then $a ⋄^{″} b := L_{a}^{'} b$ .
If $x \in N$ and $a \in S M$ , then $b ⋄^{″} a := R_{a}^{'} b$ .
If $x, y \in N$ , then $x ⋄^{″} y := x ⋄^{'} y$ .

Furthermore, the 817 law $x ⋄^{″} S S^{'} x = x$ fails for any $x \in N$ .

Proof ▶

We need to show that $⋄^{″}$ verifies the law Equation 1. In the case when $x, y \in S M$ , then the claim follows from the fact that $S M$ already obeyed this equation. If $x$ was equal to an element $a \in S M$ and $y \in N$ , then by construction the law is equivalent to $L_{S a}^{'} R_{a}^{'} L_{a}^{'} y = y$ , which follows from axiom (i).

Now suppose that $x \in N$ and $y$ is equal to some element $a$ of $S M$ . From axiom (v) we have $x ⋄^{″} x = S^{'} x$ , and then this case of Equation 1 becomes

L_{S^{'} x} (R_{a}^{'} x ⋄^{'} x) = a

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 $a \in S M$ . 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 $y ⋄^{″} x = z$ for some $z \in N$ with $z ⋄^{″} y = (L_{S^{'} y}^{'})^{- 1} x$ . But this implies $L_{S^{'} y}^{'} (z ⋄^{″} y) = x$ , which is Equation 1.

We have now verified that $⋄^{″}$ obeys 1729. For any $x \in N$ , we have $x ⋄^{″} S S^{'} x = R_{S S^{'} x}^{'} x$ , and so the final claim follows from axiom (ii).

To build a magma obeying 1729 but not 817, it thus suffices to produce

a 1729 magma $S M$ ;
a set $N$ of “non-squares”;
a squaring map $S^{'} : N \to S M$ ;
bijections $L_{a}^{'}, R_{a}^{'} : N \to N$ for all $a \in S M$ obeying the axioms (i)-(iv); and
an operation $⋄^{'} : N \times N \to S M ⊎ N$ obeying the axioms (v)-(vii).

The magma $S M$ is defined as follows:

Definition 18.2 Definition of

S M

✓

Take $S M$ to be a countably infinite abelian group of exponent $4$ , generated by generators $E_{n}$ for $n \in N$ subject to the relations $4 E_{n} = 0$ .

Lemma 18.3 Basic properties of

S M

✓

$S M$ is a 1729 magma, the squaring operation $S : S M \to S M$ is just the doubling map $S a = 2 a$ , and the double squaring map $S^{2} : S M \to S M$ is constant: $S^{2} a = 0$ for all $a \in S M$ .

Proof ▶

We now define $N$ , as well as some Cayley graph structures on it.

Definition 18.4 Definition of

N

✓

Take $N$ to be the free non-abelian group with a generator $e_{a}$ for each $a \in S M$ , thus $N$ is the set of reduced words using the alphabet $e_{a}$ , $e_{a}^{- 1}$ . Two elements $x, y \in S M$ are said to be adjacent if $x = e_{a} y$ or $y = e_{a} x$ for some $a \in S M$ ; 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 $a, b, c \in S M$ are distinct, then

1 \leq e_{c} \leq e_{b}^{- 1} e_{c} \leq e_{a} e_{b}^{- 1} e_{c} .

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 $e_{b}^{- 1} e_{c}$ .

Lemma 18.5 Basic properties of

N

✓

$N$ is countable, and $\leq$ is a partial ordering.

Proof ▶

We will define the right multiplication operators $R_{a}^{'} : N \to N$ using the group action:

Definition 18.6 Definition of

R_{a}^{'}

✓

We set

R_{a}^{'} x := e_{a} x

for all $a \in S M$ and $x \in N$ .

Lemma 18.7 Basic properties of

R_{a}^{'}

✓

The operators $R_{a}^{'}$ are bijective and obey axiom (ii).

Proof ▶

We defer construction of the squaring map $S^{'} : N \to S M$ for now, but turn to left-multiplication. From two applications of Equation 2 and the exponent 4 hypothesis we have

L_{a}^{'} = (R_{a}^{'})^{- 1} (L_{2 a}^{'})^{- 1} = (R_{a}^{'})^{- 1} L_{0}^{'} R_{2 a}^{'} .

Thus, once $L_{0}^{'}$ is specified, we can define $L_{a}^{'}$ for all other $a \in S M$ by the rule

L_{a}^{'} := (R_{a}^{'})^{- 1} L_{0}^{'} R_{2 a}^{'} .

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

Lemma 18.8 Using

L_{0}^{'}

to construct

L_{a}^{'}

✓

Suppose we have a bijection $L_{0}^{'} : N \to N$ that obeys axiom (i’), and then define $L_{a}^{'}$ for all $a \in S M$ 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 obey axiom (i). Furthermore, we have

(L_{a}^{'})^{- 1} := (R_{2 a}^{'})^{- 1} L_{0}^{'} R_{0}^{'} R_{a}^{'}

for all $a \in S M$ .

Proof ▶

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 $S M$ :

(iii’) If $R_{a}^{'} x = y$ for some $a \in S M$ and some $x, y \in N$ , then $(R_{S^{'} y}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} y}^{'} (R_{a - S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 (a - S^{'} x)}^{'} y = x$ .
(iv’) For all $x \in N$ , we have $(R_{S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} x}^{'} (R_{S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} x}^{'} x = x$ .
(v) For all $x \in N$ , we have $x ⋄^{'} x = S^{'} x$ .
(vi’) For all $y \in N$ and $a \in S M$ , we have $e_{a} y ⋄^{'} y = a - S^{'} y$ .
(vii’) For all $x, y \in N$ with $x ⋄^{'} y$ not already covered by rules (v) or (v’), we have $x ⋄^{'} y = z$ for some $z \in N$ . Furthermore, $z ⋄^{'} x = (R_{2 S^{'} x}^{'})^{- 1} L_{0}^{'} e_{0} e_{S^{'} x} y$ .

Lemma 18.9 Reduction to new axioms

✓

Suppose we can find a function $S^{'} : N \to S M$ , a bijection $L_{0}^{'} : N \to N$ , and an operation $⋄^{'} : N \times N \to S M ⊎ N$ obeying axioms (i’), (iii’), (iv’), (v), (vi’), (vii’). Then there exists a magma obeying 1729 but not 817.

Proof ▶

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 $S^{'} : N \to S M$ , a bijection $L_{0}^{'} : N \to N$ , and an operation $⋄^{'} : N \times N \to M$ obeying 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

L_{0}^{'} x = y

for some $x, y \in N$ would automatically imply that

L_{0}^{'} (R_{0}^{'})^{n} x = (R_{0}^{'})^{n} y

and also

L_{0}^{'} (R_{0}^{'})^{n} y = (R_{0}^{'})^{n - 1} x

for all $n \in Z$ . Thus, $L_{0}^{'}$ becomes defined on two right cosets $⟨ e_{0} ⟩ x$ , $⟨ e_{0} ⟩ y$ of $N$ , where $⟨ e_{0} ⟩ := {e_{0}^{n} : n \in Z}$ is an infinite cyclic subgroup of $N$ . In general, we will require that $L_{0}^{'}$ is defined on a finite union of cosets of $⟨ e_{0} ⟩$ .

In a somewhat similar vein, axiom (vii’), if iterated naively, would mean that a given entry $x ⋄^{'} y = z$ 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 “ $z ⋄^{'} x = (R_{2 S^{'} x}^{'})^{- 1} L_{0}^{'} R_{0}^{'} e_{S^{'} x} y$ ” for some $x, y, z \in N$ , which will be temporarily undefined because $S^{'} x$ is undefined. More precisely,

Definition 18.10 Partial solution

✓

A partial solution $(L_{0}^{'}, ⋄^{'}, S^{'}, I)$ 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 $⟨ e_{0} ⟩$ ;
A partially defined operation $⋄^{'} : N \times N \to M$ , defined on a finite set;
A partially defined function $S^{'} : N \to S M$ , defined on a finite set; and
A finite collection $I$ 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 “ $z ⋄^{'} x = (R_{2 S^{'} x}^{'})^{- 1} L_{0}^{'} R_{0}^{'} R_{S^{'} x}^{'} y$ ” 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 $n \in Z$ .
(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 $a \in S M$ and some $x, y \in N$ , and $S^{'} x, S^{'} y$ are defined, then $(R_{S^{'} y}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} y}^{'} (R_{a - S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 (a - S^{'} x)}^{'} y$ is defined and equal to $x$ .
(iv”) If $x \in N$ is such that $S^{'} x$ is defined, then $(R_{S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} x}^{'} (R_{S^{'} x}^{'})^{- 1} L_{0}^{'} R_{2 S^{'} x}^{'} x$ is defined and equal to $x$ .
(v”) If $x \in N$ and $x ⋄^{'} x$ is defined, then $S^{'} x$ is defined and equal to $x ⋄^{'} x$ .
(vi”) For all $y \in N$ and $a \in S M$ , if $R_{a}^{'} y ⋄^{'} y$ is defined, then $a - S^{'} y$ is defined and equal to $R_{a}^{'} y ⋄^{'} y$ .
(vii”) For all $x, y \in N$ and $x$ is not equal to $y$ or $R_{a}^{'} y$ for any $a \in S M$ , and $x ⋄^{'} y$ is defined, then it is equal to some $z \in N$ . Furthermore, either $I [x, y, z]$ is a pending identity, or else $z ⋄^{'} x$ and $(R_{2 S^{'} x}^{'})^{- 1} L_{0}^{'} R_{0}^{'} R_{S^{'} x}^{'} y$ are defined and equal to each other.
(P) If $I [x, y, z]$ is a pending identity, then $x, y, z \in N$ , and $S x$ and $z ⋄^{'} x$ are undefined. Furthermore, $z$ is not equal to $x$ or $R_{a}^{'} x$ for any $a \in S M$ , 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 $x ⋄^{'} y = z$ .
(L) If $L_{0}^{'} x$ is defined, then it is not of the form $R_{b}^{'} (R_{2 b}^{'})^{- 1} x$ for any $b \in S M$ .

We say that one partial solution $({\tilde{L}}_{0}^{'}, {\tilde{⋄}}^{'}, {\tilde{S}}^{'}, \tilde{I})$ extends another if $(L_{0}^{'}, ⋄^{'}, S^{'}, I)$ if ${\tilde{L}}^{'}$ is an extension of $L_{0}^{'}$ , ${\tilde{⋄}}^{'}$ is an extension of $⋄^{'}$ , and ${\tilde{S}}^{'}$ is an extension of $S^{'}$ . (No constraint is imposed on the final components $\tilde{I}, I$ .) This is a preordering.

Lemma 18.11 Existence of partial solution

✓

There exists a partial solution.

Proof ▶

Set $L_{0}^{'}, ⋄^{'}, S^{'}$ to be empty functions, and have the set of pending identities to also be empty. The verification of the required axioms is then routine.

Lemma 18.12 Chain of partial solutions

✓

Suppose that one has a sequence $(L_{0, n}^{'}, ⋄_{n}^{'}, S_{n}^{'}, I_{n})$ of partial solutions, each one an extension of the previous, such that for any $x, y \in N$ , $L_{0, n}^{'} x$ , $x ⋄_{n}^{'} y$ , and $S_{n}^{'} x$ are defined for some $n$ . Then there exists a 1729 magma that does not obey 817.

Proof ▶

Take the direct limit of the chain to obtain total functions $L_{0}^{'}, ⋄^{'}, S^{'}$ . The axioms (i’), (iii’), (iv”), (v”), (vi”), (vii”) of the partial solutions then easily imply that the direct limit obeys 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:

Proposition 18.13 Enlarging

L_{0}^{'}

✓

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.

Proof ▶

By axiom (i”), $L_{0}^{'} (R_{0}^{'})^{n} x$ is undefined for every integer $n$ . Let $d = E_{m}$ be a generator of $S M$ 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

L_{0}^{'} (R_{0}^{'})^{n} x := (R_{0}^{'})^{n} e_{d}

and

L_{0}^{'} (R_{0}^{'})^{n} e_{d} := (R_{0}^{'})^{n - 1} x .

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

Proposition 18.14 Enlarging

L_{0}^{'}

many times

✓

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 $x \in A$ .

Proof ▶

Next, we provide a tool for enlarging the domain of definition of $S^{'}$ . The main step is the following inductive one with extra axioms.

Proposition 18.15 Enlarging

S^{'}

with induction hypothesis and 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 $y < x$ . (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 $a \in S M$ , then $L_{0}^{'} R_{S^{'} y_{0}}^{'} x$ is defined.
(B) If $x = R_{a}^{'} y_{0}$ for some $a \in S M$ , 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.

Proof ▶

Let $d_{0}, d_{1} \in S M$ be generators $E_{n_{0}}$ of $S M$ 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 $d_{y, z}^{'} \in S M$ for $y, z \in S M$ that are distinct from each other and all previous generators (this is possible as we have infinitely many generators). We set $S^{'} x, x ⋄^{'} x := d_{0}$ ; this preserves axiom (S) and (v”). We set

L_{0}^{'} R_{2 d_{0}}^{'} x := e_{d_{1}}

L_{0}^{'} R_{2 d_{0}}^{'} (R_{d_{0}}^{'})^{- 1} e_{d_{1}} := R_{d_{0}}^{'} x

and then extend these choices using Equation 6, Equation 7 to recover axiom (i’), thus for instance

L_{0}^{'} (R_{0}^{'})^{n} R_{2 d_{0}}^{'} x := (R_{0}^{'})^{n} e_{d_{1}}

and

L_{0}^{'} (R_{0}^{'})^{n} e_{d_{1}} := (R_{0}^{'})^{n - 1} R_{2 d_{0}}^{'} x

for all $n \in Z$ .

To retain axiom (iii”), we perform the following actions:

If $y_{0}$ is undefined, do nothing.
If $R_{a}^{'} x = y_{0}$ for some $a \in S M$ (so that $L_{0}^{'} R_{S^{'} y_{0}}^{'} x$ is defined, by hypothesis (A)), set $L_{0}^{'} R_{2 (a - d_{0})}^{'} y_{0} := R_{a - d_{0}}^{'} (R_{2 S^{'} y_{0}}^{'})^{- 1} R_{0}^{'} L_{0}^{'} R_{S^{'} y_{0}}^{'} x$ . Then extend using Equation 6, Equation 7.
If $x = R_{a}^{'} y_{0}$ for some $a \in S M$ (so that $L_{0}^{'} R_{2 (a - S y_{0})}^{'} x$ is already defined, by hypothesis (B)), set $L_{0}^{'} R_{2 d_{0}}^{'} (R_{a - S^{'} y_{0}}^{'})^{- 1} L_{0}^{'} R_{2 (a - S y_{0})}^{'} x := R_{d_{0}}^{'} y_{0}$ , 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 $x^{'} ⋄ y^{'} := z^{'}$ , where $(x^{'}, y^{'}, z^{'}) := (z, x, (R_{2 d_{0}}^{'})^{- 1} e_{d_{y, z}^{'}})$ . 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 $x^{″} ⋄ y^{″} = z^{″}$ , where $(x^{″}, y^{″}, z^{″}) := (z^{'}, x^{'}, (R_{2 S^{'} x^{'}}^{'})^{- 1} L_{0}^{'} R_{0}^{'} R_{S^{'} x^{'}}^{'} y^{'})$ . 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.

Proposition 18.16 Enlarging

S^{'}

with induction hypothesis

✓

Suppose one has a partial solution in which $S^{'} x$ is undefined for some $x \in N$ , but $S^{'} y$ is defined for all $y < x$ . (This hypothesis is vacuous for $x = 1$ .) Then one can extend the partial solution so that $S^{'} x$ is now defined.

Proof ▶

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 $a \in S M$ .

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

Proposition 18.17 Enlarging

S^{'}

✓

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.

Proof ▶

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 $⋄$ :

Proposition 18.18 Enlarging

⋄

✓

Suppose one has a partial solution in which $x ⋄^{'} y$ is undefined for some $x, y \in N$ . Then one can extend the partial solution so that $x ⋄^{'} y$ is now defined.

Proof ▶

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 $x ⋄^{'} y$ is part of a pending identity). If this makes $x ⋄^{'} y$ 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 $x ⋄^{'} y := S^{'} x$ . It is clear that this preserves axiom (v”), and no other axiom is impacted.

Case 2: If $x = R_{a}^{'} y$ for some $a \in S M$ , then we set $x ⋄^{'} y := a - S^{'} y$ . 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 $a \in S M$ . Let $d_{0} \in S M$ 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 $x ⋄^{'} y := z$ with $z := e_{d_{0}}^{2}$ .

This temporarily disrupts axiom (vii”). To recover it, we perform the following actions.

Set $x^{'} ⋄^{'} y^{'} = z^{'}$ , where $(x^{'}, y^{'}, z^{'}) := (z, x, (R_{2 S^{'} x}^{'})^{- 1} L_{0} R_{0}^{'} R_{S x}^{'} y)$ .
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.

Theorem 18.19 1729 does not imply 817

✓

There exists a magma that obeys equation 1729 but not equation 817.

Proof ▶