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 $(a,b)=(\varphi ,-1)$, where $\varphi$ solves the golden ratio equation ${\varphi }^2=\varphi +1$.

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 $Z_E^2(G,M)$. 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 $Z_E^2(G,M)$. We define the $E$-cohomology $H_E^2(G,M)$ to be the quotient

Observe that if $E$ implies $E^{\prime }$, then $H_E^2(G,M)$ is a subgroup of $H_{E^{\prime }}^2(G,M)$. Thus, to refute an implication $E⟹E^{\prime }$, it suffices to locate a magma $G$ and a linear magma $M$ obeying both $E$ and $E^{\prime }$ 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 $Z_{1110}^2(G,M)$ is six-dimensional, so $H_{1110}^2(G,M)$ is two-dimensional. One can check numerically that it contains an element not in $H_{1629}^2(G,M)$, leading to a finite counterexample on the 25-element carrier $G\times M$ to the implication of 1629 from 1110.