A Generalization of a Theorem of Erné about the Number of Posets with a Fixed Antichain

We characterize finite, connected posets $P$ having a crown as retract. define multigraph $\mathfrak{F}(P)$ whose vertices are the so-called improper 4-crowns contained in extremal points of $P$, and we show that contains 4-crown retract iff there exists graph homomorphism certain type from to $\mathfrak{C}$ not depending on $P$. Additionally retract-crown with $n \geq 6$ poset induced by $n$ points.