7 problems

Let $L$ be a finite lattice, and let $A=KL$ be its incidence algebra. Assume that $A$ has a pure minimal injective coresolution. Distributivity conjecture. Then $L$ is distributive…

Let $K$ be a field of characteristic $0$, and let $P$ be a poset. Denote its incidence algebra by $KP$, and let independent and the Coxeter permutation be the corresponding notions…

Tameness conjecture for g-tame incidence algebras. If $\operatorname{Inc}_\K(P)$ is $\mathbf{g}$-tame, then $\operatorname{Inc}_\K(P)$ is tame.

Let $X$ be a finite connected poset, let $I(X,K)$ be its incidence algebra over $K$, and let $D(X,K)$ denote the diagonal subalgebra. A map $\theta:I(X,K)\to I(X,K)$ is a strong co…

Let $P$ be a finite poset, and let $\field P$ denote its incidence algebra. For the critical and regular levels $a_0,t_1,a_1,t_2,\dots,t_n,a_n$ associated with the Morse function…

Let $(X,\leqslant)$ be a locally finite pre-ordered set, let $\mathcal{R}$ be a $2$-torsion-free commutative ring with unity, and let $I(X,\mathcal{R})$ denote the incidence algebr…

Conjectured convolution-inverse formula. For the relevant indices $a,b,c,d$,