Equational Theories

18 Equation 1729

In this chapter we study magmas that obey equation 1729,

x=(yy)((yx)y).
1

for all x,y. Using the squaring operator Sy:=yy and the left and right multiplication operators Lyx:=yx and Ryx=xy, this law can be written as

LSyRyLyx=x.

This implies that Ly is injective and LSy is surjective, hence LSy is invertible. If y is a square (i.e., ySM), then Ly and LSy are both invertible, hence now Ry is also invertible, with inverse Ry1=LyLSy. We rewrite this as

Ly=Ry1LSy1
2

for all ySM.

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

Theorem 18.1 Extending a 1729 magma

Let SM be a magma obeying 1729, and let N be another set disjoint from SM, and set M:=SMN. Suppose that we have a squaring map S:NSM (which will complement the existing squaring map S:SMSM), and bijections La,Ra:NN for all aSM (which will complement the existing bijections La,Ra:SMSM coming from SM), obeying the following axioms:

  • (i) For all aSM, we have La=(Ra)1(LSa)1.

  • (ii) For all yN, the elements RayN are distinct from each other and from y as aSM varies.

  • (iii) If Rax=y for some aSM and some x,yN, then LSyLRSx1ay=x.

  • (iv) For all xN, we have (LSx)2x=x.

Suppose also that we have an operation :N×NM obeying the following axioms:

  • (v) For all xN, we have xx=Sx.

  • (vi) For all yN and aSM, we have Rayy=LSy1a.

  • (vii) For all x,yN with xy not already covered by rules (v) or (vi), we have xy=z for some zN. Furthermore, zx=(LSx)1y.

Then one can endow M with an operation :M×MM obeying 1729 defined as follows:

  • If a,bSM, then ab=ab.

  • If aSM and xN, then ab:=Lab.

  • If xN and aSM, then ba:=Rab.

  • If x,yN, then xy:=xy.

Furthermore, the 817 law xSSx=x fails for any xN.

Proof

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

  • a 1729 magma SM;

  • a set N of “non-squares”;

  • a squaring map S:NSM;

  • bijections La,Ra:NN for all aSM obeying the axioms (i)-(iv); and

  • an operation :N×NSMN obeying the axioms (v)-(vii).

The magma SM is defined as follows:

Definition 18.2 Definition of SM
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Take SM to be a countably infinite abelian group of exponent 4, generated by generators En for nN subject to the relations 4En=0.

Lemma 18.3 Basic properties of SM

SM is a 1729 magma, the squaring operation S:SMSM is just the doubling map Sa=2a, and the double squaring map S2:SMSM is constant: S2a=0 for all aSM.

Proof

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

Definition 18.4 Definition of N
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Take N to be the free non-abelian group with a generator ea for each aSM, thus N is the set of reduced words using the alphabet ea, ea1. Two elements x,ySM are said to be adjacent if x=eay or y=eax for some aSM; this defines a left Cayley graph on N. We make a partial ordering on N by declaring yx 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,cSM are distinct, then

1eceb1eceaeb1ec.

If xN is not the identity, we define the parent of x to be the unique element yN adjacent to x whose reduced word is shorter. For instance, the parent of eaeb1ec is eb1ec.

Lemma 18.5 Basic properties of N
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N is countable, and is a partial ordering.

Proof

We will define the right multiplication operators Ra:NN using the group action:

Definition 18.6 Definition of Ra
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We set

Rax:=eax
3

for all aSM and xN.

Lemma 18.7 Basic properties of Ra

The operators Ra are bijective and obey axiom (ii).

Proof

We defer construction of the squaring map S:NSM for now, but turn to left-multiplication. From two applications of Equation 2 and the exponent 4 hypothesis we have

La=(Ra)1(L2a)1=(Ra)1L0R2a.

Thus, once L0 is specified, we can define La for all other aSM by the rule

La:=(Ra)1L0R2a.
4

Furthermore, from the a=0 case of Equation 2 we must also have the axiom

  • (i’) (L0)2=(R0)1.

Conversely, we have

Lemma 18.8 Using L0 to construct La
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Suppose we have a bijection L0:NN that obeys axiom (i’), and then define La for all aSM by the formula 4. Then this recovers L0 when a=0 (to formalize this it may be convenient to give L0 and La distinct names), and the La are all bijections and obey axiom (i). Furthermore, we have

(La)1:=(R2a)1L0R0Ra
5

for all aSM.

Proof

We now write the other remaining axioms in terms of L0 rather than La using Equation 4, Equation 5, Equation 3, and the magma law on SM:

  • (iii’) If Rax=y for some aSM and some x,yN, then (RSy)1L0R2Sy(RaSx)1L0R2(aSx)y=x.

  • (iv’) For all xN, we have (RSx)1L0R2Sx(RSx)1L0R2Sxx=x.

  • (v) For all xN, we have xx=Sx.

  • (vi’) For all yN and aSM, we have eayy=aSy.

  • (vii’) For all x,yN with xy not already covered by rules (v) or (v’), we have xy=z for some zN. Furthermore, zx=(R2Sx)1L0e0eSxy.

Lemma 18.9 Reduction to new axioms

Suppose we can find a function S:NSM, a bijection L0:NN, and an operation :N×NSMN obeying axioms (i’), (iii’), (iv’), (v), (vi’), (vii’). Then there exists a magma obeying 1729 but not 817.

Proof

Our task is now to find a function S:NSM, a bijection L0:NN, and an operation :N×NM 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 L0 to be partially defined on just finitely many values: any relation of the form

L0x=y

for some x,yN would automatically imply that

L0(R0)nx=(R0)ny
6

and also

L0(R0)ny=(R0)n1x
7

for all nZ. Thus, L0 becomes defined on two right cosets e0x, e0y of N, where e0:={e0n:nZ} is an infinite cyclic subgroup of N. In general, we will require that L0 is defined on a finite union of cosets of e0.

In a somewhat similar vein, axiom (vii’), if iterated naively, would mean that a given entry xy=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 “zx=(R2Sx)1L0R0eSxy” for some x,y,zN, which will be temporarily undefined because Sx is undefined. More precisely,

Definition 18.10 Partial solution
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A partial solution (L0,,S,I) is a collection of the following data:

  • A partially defined function L0:NN, defined on a finite union of right cosets of e0;

  • A partially defined operation :N×NM, defined on a finite set;

  • A partially defined function S:NSM, 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,zN, or as formal strings of the form “zx=(R2Sx)1L0R0RSxy” for some x,y,zN.

Furthermore, the following axioms are satisfied:

  • (i”) L0x is defined and equal to y, then we have the identities Equation 6, Equation 7 for all nZ.

  • (S) If Sx is defined for some xN, then Sy is defined for all yx.

  • (iii”) If Rax=y for some aSM and some x,yN, and Sx,Sy are defined, then (RSy)1L0R2Sy(RaSx)1L0R2(aSx)y is defined and equal to x.

  • (iv”) If xN is such that Sx is defined, then (RSx)1L0R2Sx(RSx)1L0R2Sxx is defined and equal to x.

  • (v”) If xN and xx is defined, then Sx is defined and equal to xx.

  • (vi”) For all yN and aSM, if eayy is defined, then aSy is defined and equal to Rayy.

  • (vii”) For all x,yN and x is not equal to y or Ray for any aSM, and xy is defined, then it is equal to some zN. Furthermore, either I[x,y,z] is a pending identity, or else zx and (R2Sx)1L0R0RSxy are defined and equal to each other.

  • (P) If I[x,y,z] is a pending identity, then x,y,zN, and Sx and zx are undefined. Furthermore, z is not equal to x or Rax for any aSM.

  • (P’) If I[x,y,z] and I[x,y,z] are pending identities, then y=y.

We say that one partial solution (L~0,~,S~,I~) extends another if (L0,,S,I) if L~ is an extension of L0, ~ is an extension of , and S~ is an extension of S. (No constraint is imposed on the final components I~,I.) This is a preordering.

Lemma 18.11 Existence of partial solution
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There exists a partial solution.

Proof
Lemma 18.12 Chain of partial solutions
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Suppose that one has a sequence (L0,n,n,Sn,In) of partial solutions, each one an extension of the previous, such that for any x,yN, L0,nx, xny, and Snx are defined for some n. Then there exists a 1729 magma that does not obey 817.

Proof

Now we seek to enlarge a partial solution. We first make an easy observation:

Proposition 18.13 Enlarging L0
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Suppose one has a partial solution in which L0x is undefined for some xN. Then one can extend the partial solution so that L0x is now defined.

Proof

As a corollary, we have

Proposition 18.14 Enlarging L0 many times
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Suppose one has a partial solution. Let A be a finite subset of N. Then one can extend the partial solution so that L0x is now defined for all xA.

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
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Suppose one has a partial solution in which Sx is undefined for some xN, but Sy is defined for all y<x. (This hypothesis is vacuous for x=1.) Let y0 be the parent of x (if x1), and assume the following additional axioms:

  • (A) If Rax=y0 for some aSM, then L0RSy0x is defined.

  • (B) If x=Ray0 for some aSM, then L0R2(aSy0)x is defined.

  • (C) If I[x,y,z] for some y,zN, and Sz is defined, then L0R0RSzx is defined.

Then one can extend the partial solution so that Sx is now defined.

Proof
Proposition 18.16 Enlarging S with induction hypothesis
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Suppose one has a partial solution in which Sx is undefined for some xN, but Sy is defined for all y<x. (This hypothesis is vacuous for x=1.) Then one can extend the partial solution so that Sx is now defined.

Proof

As a corollary, we have

Proposition 18.17 Enlarging S
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Suppose one has a partial solution in which Sx is undefined for some xN. Then one can extend the partial solution so that Sx is now defined.

Proof

Finally, we give a tool for enlarging :

Proposition 18.18 Enlarging
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Suppose one has a partial solution in which xy is undefined for some x,yN. Then one can extend the partial solution so that xy is now defined.

Proof
Theorem 18.19 1729 does not imply 817
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There exists a magma that obeys equation 1729 but not equation 817.

Proof