Weakly Cohen-Macaulay posets and a class of finite-dimensional graded quadratic algebras

To a finite ranked poset $\Gamma$ we associate a finite-dimensional graded quadratic algebra $R_\Gamma$. Assuming $\Gamma$ satisfies a combinatorial condition known as uniform, $R_{\Gamma}$ is related to a well-known algebra, the splitting algebra $A_{\Gamma}$. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset $\Gamma$, we ask: Is $R_{\Gamma}$ Koszul? The Koszulity of $R_{\Gamma}$ is related to a combinatorial topology property of $\Gamma$ called Cohen-Macaulay. Kloefkorn and Shelton proved that if $\Gamma$ is a finite ranked cyclic poset, then $\Gamma$ is Cohen-Macaulay if and only if $\Gamma$ is uniform and $R_{\Gamma}$ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay, and we note that this new class includes posets with disconnected open subintervals. We prove: if $\Gamma$ is a finite ranked cyclic poset, then $\Gamma$ is weakly Cohen-Macaulay if and only if $R_{\Gamma}$ is Koszul.