Splitting finite antichains in the homomorphism order

A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order.