  We study maximum antichains in two posets related to quiver representations.
Firstly, we consider the set of isomorphism classes of indecomposable
representations ordered by inclusion. For various orientations of the Dynkin
diagram of type A we construct a maximum antichain in the poset. Secondly, we
consider the set of subrepresentations of a given quiver representation, again
ordered by inclusion. It is a finite set if we restrict to linear
representations over finite fields or to representations with values in the
category of pointed sets. For particular situations we prove that this poset is
Sperner.
