18 Equation 1729
In this chapter we study magmas that obey equation 1729,
for all . Using the squaring operator and the left and right multiplication operators and , this law can be written as
This implies that is injective and is surjective, hence is invertible. If is a square (i.e., ), then and are both invertible, hence now is also invertible, with inverse . We rewrite this as
for all .
We have the following procedure for extending a small magma obeying Equation 1 to a larger one :
Theorem
18.1
Extending a 1729 magma
Let be a magma obeying 1729, and let be another set disjoint from , and set . Suppose that we have a squaring map (which will complement the existing squaring map ), and bijections for all (which will complement the existing bijections coming from ), obeying the following axioms:
(i) For all , we have .
(ii) For all , the elements are distinct from each other and from as varies.
(iii) If for some and some , then .
(iv) For all , we have .
Suppose also that we have an operation obeying the following axioms:
(v) For all , we have .
(vi) For all and , we have .
(vii) For all with not already covered by rules (v) or (vi), we have for some . Furthermore, .
Then one can endow with an operation obeying 1729 defined as follows:
If , then .
If and , then .
If and , then .
If , then .
Furthermore, the 817 law fails for any .
Proof
▶
We need to show that verifies the law Equation 1. In the case when , then the claim follows from the fact that already obeyed this equation. If was equal to an element and , then by construction the law is equivalent to , which follows from axiom (i).
Now suppose that and is equal to some element of . From axiom (v) we have , and then this case of Equation 1 becomes
which follows from axiom (vi). So the only remaining case is when . Using axiom (ii), we can divide into cases:
Case 1: . Then by (v) we need to show that , which follows from axiom (iv).
Case 2: for some . Then by axiom (vi), we need to show that , which follows from axiom (iii).
Case 3: We are not in case 1 or case 2. Then by axiom (vii), we have for some with . But this implies , which is Equation 1.
We have now verified that obeys 1729. For any , we have , and so the final claim follows from axiom (ii).
To build a magma obeying 1729 but not 817, it thus suffices to produce
The magma is defined as follows:
Definition
18.2
Definition of
Take to be a countably infinite abelian group of exponent , generated by generators for subject to the relations .
Lemma
18.3
Basic properties of
is a 1729 magma, the squaring operation is just the doubling map , and the double squaring map is constant: for all .
We now define , as well as some Cayley graph structures on it.
Definition
18.4
Definition of
Take to be the free non-abelian group with a generator for each , thus is the set of reduced words using the alphabet , . Two elements are said to be adjacent if or for some ; this defines a left Cayley graph on . We make a partial ordering on by declaring if is a right subword of (or equivalently, is on the unique simple path from to ). For instance, if are distinct, then
If is not the identity, we define the parent of to be the unique element adjacent to whose reduced word is shorter. For instance, the parent of is .
Lemma
18.5
Basic properties of
is countable, and is a partial ordering.
We will define the right multiplication operators using the group action:
Definition
18.6
Definition of
Lemma
18.7
Basic properties of
The operators are bijective and obey axiom (ii).
We defer construction of the squaring map for now, but turn to left-multiplication. From two applications of Equation 2 and the exponent 4 hypothesis we have
Thus, once is specified, we can define for all other by the rule
Furthermore, from the case of Equation 2 we must also have the axiom
Conversely, we have
Lemma
18.8
Using to construct
Suppose we have a bijection that obeys axiom (i’), and then define for all by the formula 4. Then this recovers when (to formalize this it may be convenient to give and distinct names), and the are all bijections and obey axiom (i). Furthermore, we have
for all .
We now write the other remaining axioms in terms of rather than using Equation 4, Equation 5, Equation 3, and the magma law on :
(iii’) If for some and some , then .
(iv’) For all , we have .
(v) For all , we have .
(vi’) For all and , we have .
(vii’) For all with not already covered by rules (v) or (v’), we have for some . Furthermore, .
Lemma
18.9
Reduction to new axioms
Suppose we can find a function , a bijection , and an operation obeying axioms (i’), (iii’), (iv’), (v), (vi’), (vii’). Then there exists a magma obeying 1729 but not 817.
Our task is now to find a function , a bijection , and an operation 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 to be partially defined on just finitely many values: any relation of the form
for some would automatically imply that
and also
for all . Thus, becomes defined on two right cosets , of , where is an infinite cyclic subgroup of . In general, we will require that is defined on a finite union of cosets of .
In a somewhat similar vein, axiom (vii’), if iterated naively, would mean that a given entry 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 of the form “” for some , which will be temporarily undefined because is undefined. More precisely,
Definition
18.10
Partial solution
A partial solution is a collection of the following data:
A partially defined function , defined on a finite union of right cosets of ;
A partially defined operation , defined on a finite set;
A partially defined function , defined on a finite set; and
A finite collection of “pending identities” , which one can think of either as ordered triples of elements , or as formal strings of the form “” for some .
Furthermore, the following axioms are satisfied:
(i”) is defined and equal to , then we have the identities Equation 6, Equation 7 for all .
(S) If is defined for some , then is defined for all .
(iii”) If for some and some , and are defined, then is defined and equal to .
(iv”) If is such that is defined, then is defined and equal to .
(v”) If and is defined, then is defined and equal to .
(vi”) For all and , if is defined, then is defined and equal to .
(vii”) For all and is not equal to or for any , and is defined, then it is equal to some . Furthermore, either is a pending identity, or else and are defined and equal to each other.
(P) If is a pending identity, then , and and are undefined. Furthermore, is not equal to or for any .
(P’) If and are pending identities, then .
We say that one partial solution extends another if if is an extension of , is an extension of , and is an extension of . (No constraint is imposed on the final components .) This is a preordering.
Lemma
18.11
Existence of partial solution
There exists a partial solution.
Proof
▶
Set 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 of partial solutions, each one an extension of the previous, such that for any , , , and are defined for some . Then there exists a 1729 magma that does not obey 817.
Proof
▶
Take the direct limit of the chain to obtain total functions . 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
Suppose one has a partial solution in which is undefined for some . Then one can extend the partial solution so that is now defined.
Proof
▶
By axiom (i”), is undefined for every integer . Let be a generator of that does not appear as a component of any index of any of the generators appearing anywhere in the partial solution; such a exists due to the finiteness hypotheses. We set , and then extend by Equation 6, Equation 7, thus
and
Because of the new nature of , no collisions in the partial function 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’) are affected by this extension.
As a corollary, we have
Proposition
18.14
Enlarging many times
Suppose one has a partial solution. Let be a finite subset of . Then one can extend the partial solution so that is now defined for all .
Proof
▶
Iterate Proposition 18.13 in the obvious fashion.
Next, we provide a tool for enlarging the domain of definition of . The main step is the following inductive one with extra axioms.
Proposition
18.15
Enlarging with induction hypothesis and axioms
Suppose one has a partial solution in which is undefined for some , but is defined for all . (This hypothesis is vacuous for .) Let be the parent of (if ), and assume the following additional axioms:
(A) If for some , then is defined.
(B) If for some , then is defined.
(C) If for some , and is defined, then is defined.
Then one can extend the partial solution so that is now defined.
Proof
▶
Let be distinct generators of that do not appear in the index of any that currently appears in the partial solution (of which there are only finitely many). We also set some further distinct generators for that are distinct from each other and all previous generators (this is possible as we have infinitely many generators). We set ; this preserves axiom (S) and (v”). In order to retain axiom (iv”), we now also set
and then extend these choices using Equation 6, Equation 7 to recover axiom (i’), thus for instance
and
for all . To retain axiom (iii”), we perform the following actions:
If is undefined, do nothing.
If for some (so that is defined, by hypothesis (A)), set . Then extend using Equation 6, Equation 7.
If for some (so that is already defined, by hypothesis (B)), set , and extend using Equation 6, Equation 7.
To retain axiom (P), we perform the following actions for each pending identity of the form for some , executed in some arbitrary order.
Remove this identity from the list of pending identities.
Set , and , where . Extend all the new definitions of using Equation 6, Equation 7.
If is undefined, add as a pending identity.
If instead is defined (which makes defined, by axiom (C)), set , where . Then add as a pending identity.
One can check that these definitions do not cause any collisions in the partial function , and that axioms (i’), (iii’), (iv”), (v”), (vii”), (P), (P’) are preserved; the remaining axiom (vi”) is unaffected by this extension.
Proposition
18.16
Enlarging with induction hypothesis
Suppose one has a partial solution in which is undefined for some , but is defined for all . (This hypothesis is vacuous for .) Then one can extend the partial solution so that is now defined.
Proof
▶
Let be the parent of , that is to say the unique neighbor of in the path to (this is only defined for ), then by axiom (i”) is the unique neighbor for which is defined, and we either have or for some unique .
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
Suppose one has a partial solution in which is undefined for some . Then one can extend the partial solution so that is now defined.
Proof
▶
Obtained by induction from Proposition 18.16, using the fact that there are no infinite descending chains in .
Finally, we give a tool for enlarging :
Proposition
18.18
Enlarging
Suppose one has a partial solution in which is undefined for some . Then one can extend the partial solution so that is now defined.
Proof
▶
By applying Proposition 18.17 as needed, we may assume without loss of generality that and are already defined (among other things, this removes the possibility that is part of a pending identity). If this makes defined, then we are done, so we may assume that this is not the case.
We now divide into cases:
Case 1: . In this case we set . It is clear that this preserves axiom (v”), and no other axiom is impacted.
Case 2: If for some , then we set . This preserves axiom (vi”), and no other axiom is impacted.
Case 3: If is not equal to or for any . Let be generators that do not appear as a component of any index of any of the generators appearing anywhere in the partial solution. We set with .
This temporarily disrupts axiom (vii”). To recover it, we perform the following actions.
If is not currently defined, we set it equal to , and extend by Equation 6, Equation 7.
Set , where .
Add as a pending identity.
One can check that these definitions do not cause any collisions in the partial function , 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.