11 Magma cohomology

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

where \(C^n(G,M)\) is the space of functions \(f: G^n \to M\), and the coboundary maps \(d: C^n(G,M) \to C^{n+1}(G,M)\) are explicit maps obeying the relation \(d^2=0\). The cohomology group \(H^2(G,M)\) 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 \(w_{E,1}(x_1,\dots ,x_n) = w_{E,2}(x_1,\dots ,x_n)\). We let \(M = (M,+)\) be an abelian group that also has a linear magma operation

obeying \(E\) for some \(a,b \in \mathrm{End}(M)\) in the endomorphism ring of \(M\). Note that in such a linear magma, any word \(w(s_1,\dots ,s_n)\) takes the form

for some (non-commutative) polynomials \(P_{w,i}(a,b)\) in \(a,b\) with natural number coefficients. For instance,

The law \(E\) is then obeyed when

for \(i=1,\dots ,n\). For instance, the law 1110,

would be obeyed if one has

A solution here would be provided by \((a,b) = (\phi ,-1)\), where \(\phi \) solves the golden ratio equation \(\phi ^2 = \phi + 1\).

We consider extensions of \(G\) by \(M\), which are magmas with carrier \(G \times M\) with an operation of the form

for some function \(f: G \times G \to M\). An easy induction then shows that for any word \(w(x_1,\dots ,w_n)\), one has

where \(w_1 \diamond w_2\) ranges over all subterms of \(w\) that are not single variables, and \(Q_{w,w_1 \diamond w_2}(a,b)\) is a suitable (noncommutative) monomial in \(a,b\). For instance

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

for all \(x,y \in G\). More generally, an extension obeys a law \(E\) provided that

for all \(x_1,\dots ,x_n \in G\). We call \(f\) a \(E\)-cocycle if this equation holds, and denote the space of such \(E\)-cocycles as \(Z^2_E(G,M)\). This is an abelian group, and each \(E\)-cocycle defines a magma on \(G \times M\) obeying \(E\). For instance, when \(f=0\) we obtain the direct product of the \(G\) and \(M\) magmas.

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

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 \(f: G \times G \to M\) of the form \(f(x,y) =g(x \diamond y) - ag(x) - bg(y)\) for some \(g: G \to M\), we can add a coboundary to an \(E\)-cocycle and still obtain a \(E\)-cocycle. So if we let \(B^2(G,M)\) be the space of coboundaries, we see that \(B^2(G,M)\) is a subgroup of \(Z^2_E(G,M)\). We define the \(E\)-cohomology \(H^2_E(G,M)\) to be the quotient

Observe that if \(E\) implies \(E'\), then \(H^2_E(G,M)\) is a subgroup of \(H^2_{E'}(G,M)\). Thus, to refute an implication \(E \implies E'\), it suffices to locate a magma \(G\) and a linear magma \(M\) obeying both \(E\) and \(E'\) such that

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:

A simultaneous (linear) model for both Equation 1 and Equation 6 is given by carrier \(G=M={\mathbb F}_5\) with \(x \diamond y = 3x-y\). Then the coboundaries \(f: {\mathbb F}_5 \times {\mathbb F}_5 \to {\mathbb F}_5\) are of the form \(f(x,y) = g(3x-y) - 3g(x) + g(y)\) for \(g: {\mathbb F}_5 \to {\mathbb F}_5\), the \(1110\)-cocycles solve the equation

for \(x,y \in {\mathbb F}_5\), and the \(1629\)-cocycles solve the equation

A function \(g:{\mathbb F}_5 \to {\mathbb F}_5\) has vanishing derivative, \(g(3x-y) - 3g(x) + g(y) = 0\), if and only if \(g\) is linear, which is a one-dimensional space; so, by the rank-nullity theorem, the space \(B^2(G,M)\) of coboundaries is four-dimensional. One can check computationally that the space \(Z^2_{1110}(G,M)\) is six-dimensional, so \(H^2_{1110}(G,M)\) is two-dimensional. One can check numerically that it contains an element not in \(H^2_{1629}(G,M)\), leading to a finite counterexample on the 25-element carrier \(G \times M\) to the implication of 1629 from 1110.