Measures on Partial Orders

For the Fraïssé class $\mathfrak P$ of finite partially ordered sets, every measure $\nu$ is completely determined by the nine values $$\nu_{p,q}=\nu(J_{p,q},*),\qquad 0\le p,q\le2.$$ The possible $3\times3$ matrices $(\nu_{p,q})$ are classified explicitly. Consequently, the reduced space of all measures is exactly the union of one plane, eight lines, and 15 isolated points, with the plane and three lines meeting at one point and no other intersections. The paper also completely classifies weak measures and supports, proves there is no regular measure on all finite posets, classifies characteristic-zero measures for the automorphism group of the universal homogeneous poset, and determines the Knop-like measures.

For the Fraïssé class $\mathfrak P$ of finite partially ordered sets, every measure $\nu$ is completely determined by the nine values $$\nu_{p,q}=\nu(J_{p,q},*),\qquad 0\le p,q\le2.$$ The possible $3\times3$ matrices $(\nu_{p,q})$ are classified explicitly. Consequently, the reduced space of all measures is exactly the union of one plane, eight lines, and 15 isolated points, with the plane and three lines meeting at one point and no other intersections. The paper also completely classifies weak measures and supports, proves there is no regular measure on all finite posets, classifies characteristic-zero measures for the automorphism group of the universal homogeneous poset, and determines the Knop-like measures.