Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition? $\sim_{M,\mathscr{F}}\,=\bigl\{(x,y)\in M\times M\bigm|\forall A\in\mathscr{F}\,(x\in A\leftrightarrow y\in A)\bigr\}$.

The first question is about group algebras: Question 1: Let $A=kG$ be a group algebra and let $M$ be an indecomposable $A$-module. Does $Ext_A^1(M,M) \neq 0$ imply $Ext_A^2(M,M) \neq 0$ or even $Ext_A^i(M,M) \neq 0$ for all $i>0$? This is true in case $A$ is representation-finite. The next ...

Specifically, consider expressions involving integers, addition, multiplication, division, and nth roots for any positive integer n. Is there an algorithm that can determine whether such an expression is a rational number? If the expression is a rational number, is it possible to determine which ...

It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.) Now fix an arbitrary measurable space $(\Omega,\mathcal{F})$, and let $(f_\omega)_{\omega \in \Omega}$ be a fami...