On sperner's theorem
On sperner's theorem

Summary/Abstract: Let R be a finite chain ring with |R|=q2, R/radR=Fq. Every submodule M of RRn is isomorphic to a direct sum of cyclic modules:M=R/Nλ1 + R/Nλ2 + ... + R/Nλn,where λ1≥ λ2≥…≥ λn≥0 are integers and N=rad R. The n-tuple (λ1,λ2,…,λn) is called the shape of M. We consider the partially ordered set P of all submodules contained in a module of shape (λ1,λ2,…,λn). We prove an anlogue of Sperner's theorem saying that the size of a maximal antichain in P is equal to Σ[λμ]μ≤λwhere the summand is defined as the number of all modules of shape μ contained in a module of shape λ and the the sum runs over all partitions μ=(μ1,μ2,…,μn) with Σμi=[(Σiλi)/2].

• Details
• Contents