11 Magma cohomology
Group cohomology is a theory that constructs certain abelian groups to a group acting as a module on an abelian group , via a chain complex
where is the space of functions , and the coboundary maps are explicit maps obeying the relation . The cohomology group is of particular relevance in constructing extensions of the group by .
It turns out that part of this formalism can be extended to more general magmas that obey a different law than the associative law.
Let be a magma obeying some equation of the form . We let be an abelian group that also has a linear magma operation
obeying for some in the endomorphism ring of . Note that in such a linear magma, any word takes the form
for some (non-commutative) polynomials in with natural number coefficients. For instance,
The law is then obeyed when
for . For instance, the law 1110,
would be obeyed if one has
A solution here would be provided by , where solves the golden ratio equation .
We consider extensions of by , which are magmas with carrier with an operation of the form
for some function . An easy induction then shows that for any word , one has
where ranges over all subterms of that are not single variables, and is a suitable (noncommutative) monomial in . For instance
Assuming Equation 2, we conclude that this extension obeys the law Equation 1 if and only if one has
for all . More generally, an extension obeys a law provided that
for all . We call a -cocycle if this equation holds, and denote the space of such -cocycles as . This is an abelian group, and each -cocycle defines a magma on obeying . For instance, when we obtain the direct product of the and magmas.
Given any function , one can define a bijection on , which conjugates the law Equation 3 to the law
Being conjugate, this new operation will obey if and only if the original operation does. Thus if one defines a coboundary to be a function of the form for some , we can add a coboundary to an -cocycle and still obtain a -cocycle. So if we let be the space of coboundaries, we see that is a subgroup of . We define the -cohomology to be the quotient
Observe that if implies , then is a subgroup of . Thus, to refute an implication , it suffices to locate a magma and a linear magma obeying both and 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 with . Then the coboundaries are of the form for , the -cocycles solve the equation
for , and the -cocycles solve the equation
A function has vanishing derivative, , if and only if is linear, which is a one-dimensional space; so, by the rank-nullity theorem, the space of coboundaries is four-dimensional. One can check computationally that the space is six-dimensional, so is two-dimensional. One can check numerically that it contains an element not in , leading to a finite counterexample on the 25-element carrier to the implication of 1629 from 1110.