Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations

A class of stratified posets ${\displaystyle I{\cdot }_{\varrho }}$ is investigated and their incidence algebras ${\displaystyle KI{\cdot }_{\varrho }}$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on ${\displaystyle I{\cdot }_{\varrho }}$ we associate with ${\displaystyle I{\cdot }_{\varrho }}$ a bound quiver (Q, Ω) in such a way that ${\displaystyle KI{\cdot }_{\varrho }\simeq K(Q,\Omega )}$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if ${\displaystyle I{\cdot }_{\varrho }}$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition ${\displaystyle I_{\varrho }}$ is three-partite a fundamental domain ${\displaystyle I^{\cdot +\times }}$ of this covering is constructed and a functorial connection between ${\displaystyle {mod}_{sp}(K{I}^{\cdot +\times }\_\varrho )}$ and ${\displaystyle {mod}_{sp}\left(KI{\cdot }_{\varrho }\right)}$ is given.