category-theory

A general equality problem for Abelian groups Consider the problem of deciding whether of deciding whether an equation, such as $$ (x + y) + z + -(x + z) = y$$ is true in all Abelian groups. This means that for any Abelian group $A$, and for any chosen values $x$, $y$ and $z$ in $A$, the equation should be true. This hardly poses a problem, for humans atleast - by "expanding", "cancelling", "rearranging" and "grouping", anyone well-versed in high-school algebra can solve an arbitrary instance of this problem without much thought....

Introduction Mathematical objects can often be described in several ways, each description usually offering a different perspective and serving a different purpose. For example, group rings - the subject of this post - can be described in two very different ways. Given a group G and a ring R, one way to think of the group ring R[G] is as the space of functions from G to R having finite support, with addition being pointwise and multiplication given by convolution....