Equational Theories

11 Magma cohomology

Group cohomology is a theory that constructs certain abelian groups Hn(G,M) to a group G acting as a module on an abelian group M, via a chain complex

0C0(G,M)dC1(G,M)dC2(G,M)dC3(G,M)d

where Cn(G,M) is the space of functions f:GnM, and the coboundary maps d:Cn(G,M)Cn+1(G,M) are explicit maps obeying the relation d2=0. The cohomology group H2(G,M) is of particular relevance in constructing extensions of the group G by M.

It turns out that part of this formalism can be extended to more general magmas G that obey a different law than the associative law.

Let G be a magma obeying some equation E of the form wE,1(x1,,xn)=wE,2(x1,,xn). We let M=(M,+) be an abelian group that also has a linear magma operation

st=as+bt

obeying E for some a,bEnd(M) in the endomorphism ring of M. Note that in such a linear magma, any word w(s1,,sn) takes the form

w(s1,,sn)=i=1nPw,i(a,b)si

for some (non-commutative) polynomials Pw,i(a,b) in a,b with natural number coefficients. For instance,

ss=(a+b)s
t(ss)=(ba+b2)s+at
(t(ss))t=(aba+ab2)s+(a2+b)t
t((t(ss))t)=(baba+bab2)s+(ba2+b2+a)t.

The law E is then obeyed when

PwE,1,i(a,b)=PwE,2,i(a,b)

for i=1,,n. For instance, the law 1110,

x=y((y(xx))y)
1

would be obeyed if one has

baba+bab2=1;ba2+b2+a=0.
2

A solution here would be provided by (a,b)=(ϕ,1), where ϕ solves the golden ratio equation ϕ2=ϕ+1.

We consider extensions of G by M, which are magmas with carrier G×M with an operation of the form

(x,s)(y,t)=(xy,as+bt+f(x,y))
3

for some function f:G×GM. An easy induction then shows that for any word w(x1,,wn), one has

w((x1,s1),,(xn,sn))=(w(x1,,xn),i=1nPw,i(a,b)xi
+w1w2wQw,w1w2(a,b)f(w1(x1,,xn),w2(x1,,xn)))

where w1w2 ranges over all subterms of w that are not single variables, and Qw,w1w2(a,b) is a suitable (noncommutative) monomial in a,b. For instance

(x,s)(x,s)=(xx,(a+b)s+f(x,x))
(y,t)((x,s)(x,s))=(y(xx),(ba+b2)s+at+bf(x,x)+f(y,xx))
((y,t)((x,s)(x,s)))(yt)=((y(xx))y,
(aba+ab2)s+(a2+b)t+abf(x,x)+af(y,xx)+f(y(xx),y))
(yt)(((y,t)((x,s)(x,s)))(yt))=(y((y(xx))y),
(baba+bab2)s+(ba2+b2+a)t+babf(x,x)+baf(y,xx)+bf(y(xx),y)+f(y,(y(xx))y)).

Assuming Equation 2, we conclude that this extension obeys the law Equation 1 if and only if one has

babf(x,x)+baf(y,xx)+bf(y(xx),y)+f(y,(y(xx))y)=0
4

for all x,yG. More generally, an extension obeys a law E provided that

w1w2wE,1QwE,1,w1w2(a,b)f(w1(x1,,xn),w2(x1,,xn))
=w1w2wE,2QwE,2,w1w2(a,b)f(w1(x1,,xn),w2(x1,,xn))

for all x1,,xnG. We call f a E-cocycle if this equation holds, and denote the space of such E-cocycles as ZE2(G,M). This is an abelian group, and each E-cocycle defines a magma on G×M obeying E. For instance, when f=0 we obtain the direct product of the G and M magmas.

Given any function g:GM, one can define a bijection (x,s)(x,s+g(x)) on G×M, which conjugates the law Equation 3 to the law

(x,s)(y,t)=(xy,as+bt+f(x,y)+g(xy)ag(x)bg(y)).

Being conjugate, this new operation will obey E if and only if the original operation does. Thus if one defines a coboundary to be a function f:G×GM of the form f(x,y)=g(xy)ag(x)bg(y) for some g:GM, we can add a coboundary to an E-cocycle and still obtain a E-cocycle. So if we let B2(G,M) be the space of coboundaries, we see that B2(G,M) is a subgroup of ZE2(G,M). We define the E-cohomology HE2(G,M) to be the quotient

HE2(G,M):=ZE2(G,M)/B2(G,M).

Observe that if E implies E, then HE2(G,M) is a subgroup of HE2(G,M). Thus, to refute an implication EE, it suffices to locate a magma G and a linear magma M obeying both E and E such that

HE2(G,M)HE2(G,M).
5

This leads to a computational approach to refutations, as these groups can be computed by linear algebra.

For instance, let us consider the law Equation 1 together with a putative consequence, equation 1629:

x=(xx)((xx)x).
6

A simultaneous (linear) model for both Equation 1 and Equation 6 is given by carrier G=M=F5 with xy=3xy. Then the coboundaries f:F5×F5F5 are of the form f(x,y)=g(3xy)3g(x)+g(y) for g:F5F5, the 1110-cocycles solve the equation

3f(x,x)3f(y,2x)f(3y2x,y)+f(y,3yx)=0

for x,yF5, and the 1629-cocycles solve the equation

f(2x,0)f(2x,x)=0.

A function g:F5F5 has vanishing derivative, g(3xy)3g(x)+g(y)=0, if and only if g is linear, which is a one-dimensional space; so, by the rank-nullity theorem, the space B2(G,M) of coboundaries is four-dimensional. One can check computationally that the space Z11102(G,M) is six-dimensional, so H11102(G,M) is two-dimensional. One can check numerically that it contains an element not in H16292(G,M), leading to a finite counterexample on the 25-element carrier G×M to the implication of 1629 from 1110.

Remark 4
#

In the case the associative law (and taking a=b=1), the cocycle law becomes the familiar

f(x,y)+f(xy,z)=f(x,yz)+f(y,z)

and we recover the usual group cohomology (if G is a group). The same occurs for Tarski’s law 543 x=y(z(x(yz))).

Remark 5
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One can interpret the above cohomology group in terms of a partial chain complex

0C0(G,M)d0C1(G,M)d1C2(G,M)dE2Cn(G,M)

where the zeroth coboundary map d0:C0(G,M)C1(G,M) is the zero map, the first coboundary map d:C1(G,M)C2(G,M) (which does not depend on the equation E) is defined by the formula

d1vf(x,y):=f(xy)(f(x)f(y))=f(xy)af(x)bf(y)

and the second coboundary map dE2:C2(G,M)Cn(G,M) (which does depend on E) is defined by the formula

dE2f(x1,,xn):=w1w2wE,1QwE,1,w1w2(a,b)f(w1(x1,,xn),w2(x1,,xn))
w1w2wE,2QwE,2,w1w2(a,b)f(w1(x1,,xn),w2(x1,,xn)).

The fact that coboundaries are cocycles can then be rewritten as the chain complex relations d1d0=0, dE2d1=0. The group HE2(G,M) is then just the second cohomology group of this chain complex. The first cohomology group HE1(G,M) is the kernel of d1, or equivalently the abelian group of magma homomorphisms from the magma G to the linear magma M.

When E is the associative law, this partial chain complex can be extended to the usual group cohomology chain complex; however, it is not clear if any such extension exists for a general law E.

Remark 6
#

Suppose that G=M=Fp is a field of prime order, and a,b are coefficients in that field, with the magma operation on G also given in a linear form xy=ax+by. Then one can view a cocycle f:Fp×FpFp as a bivariate polynomial of degree at most 2p2 with coefficients in Fp. The coboundary maps d,dE preserve degree, and so one can decompose (or “grade”) the cohomology group HE2(G,M) as d=02p2HE2(G,M)d, where HE2(G,M)d are defined as with HE2(G,M) but with the cocycles and coboundaries required to be homogeneous polynomials of degree at most d. To disprove an implication EE, it thus suffices to establish a non-inclusion HE2(G,M)dHE2(G,M)d at a single degree d, which may be slightly easier computationally, and also provides for more compactly described counterexamples.