3 problems

Let $M$ be a finite monoid, and let $k$ be a field. Write $kM$ for the monoid algebra of $M$ over $k$, and assume that the Cartan matrix of the monoid algebra $\mathbb C M$ is non-…

Let $k$ be a commutative ring, let $M$ be an abelian monoid, and let $\mathfrak c=(c_1,c_2,\ldots)$ be a sequence of integers with $c_i\ge 2$. For each $c\ge 2$, the dilation…

Let $\Bbbk$ be an algebraically closed field of arbitrary characteristic, let $Q$ be a finitely generated commutative monoid, and let $\Bbbk[Q]$ be its monoid algebra. Mesoprimary…