Is there a diagram invariant? (Q1115889)

Given two posets P and P' the diagram of P' is said to be a reorientation of the diagram of P if P and P' have the same covering graph. A property \(\rho\) about posets is called a diagram invariant if, for every poset P which satisfies \(\rho\), every reorientation of P also satisfies \(\rho\). The authors conjecture: There is no nontrivial diagram invariant at all. The principal result of the paper is intended as partial support for it: Let \(\rho\) be a property about nonempty, finite posets. If \(\rho\) is a diagram invariant which is closed under finite direct products and retracts then \(\rho\) holds exclusively for one of these classes: (i) all finite posets, (ii) all finite, connected posets, (iii) all finite antichains, (iv) the one-element poset. The proof depends on the following interesting theorem: Every finite, connected poset is an order retract of a reorientation of a finite lattice.