A retract characterization of posets with the fixed-point property

The fixed point property of a poset and the fixed point property of the poset induced by its extremal points

“…We take a leaf L ∈ K(P ). P | L has the fixed point property according to (3). Moreover, G(P \ L) = G(P ) \ {L} fulfills (1)-( 3) and has the fixed point property according to the induction hypothesis.…”

“…We call a crown being a retract a retract-crown, and obviously, a poset P containing a retract-crown does not have the fixed point property (for a generalization, see Brualdi and Dias da Silva [3]). However, even if a crown has been found in a poset, the crucial step remains to decide if it is a retract or not.…”

“…Indeed, crowns are the "natural enemies" of the fixed point property of posets, because a crown being a retract of a poset (a retract-crown) directly delivers a fixed point free homomorphism. Brualdi and Dias da Silva [3] generalized this observation in 1997 by showing that P does not have the fixed point property iff it contains a generalized crown as retract.…”

Crowns as retracts

“…Crowns can be regarded as the "natural enemies" of the fixed point property, because a crown being a retract of a poset directly delivers a fixed point free homomorphism. Brualdi and Dias da Silva [2] generalized this observation by showing that P does not have the fixed point property iff it contains a generalized crown as retract.…”