Simetrik zincir ayrışmaları

Let $ P_n $ be the quotient poset of the action of the dihedral group $ D_n $ on the Boolean lattice $2^n$. In \cite{duffus2015symmetric}, Duffus and Thayer ask whether the poset $ P_n $ has a symmetric saturated chain decomposition. In this thesis, we give an affirmative answer to this problem for small dimensions. For this, we first introduce the notion of a symmetric saturated chain decomposition of a partially ordered set. Then we discuss some properties of partially ordered sets with symmetric saturated chain decompositions. More precisely, we show that if the posets $ P $ and $ Q $ have a symmetric saturated chain decomposition, so does the direct product $P\times Q$. Then we recall the Sperner property and show that if a poset has a symmetric saturated chain decomposition, then it has the Sperner property. We show that each division poset has a symmetric saturated chain decomposition. We also give symmetric saturated chain decompositions for $ P_n $ when $ n=5,6,7,8 $. We conclude the thesis by giving an isomorphism between the poset $ P_n $ and the quotient poset $ \tilde{Q}_n/D_n $ where $ \tilde{Q}_n $ is the poset obtained by adding a minimal element to a poset of ordered partitions of a positive integer $n$. This isomorphism might be useful to show that $ P_n $ has a symmetric saturated chain decomposition in general.