Symmetric Chain Decompositions of Products of Posets with Long Chains

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.

Symmetric Chain Decomposition of Necklace Posets

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors.

Spherical Venn Diagrams with Involutory Isometries

In this paper we give a construction, for any $n$, of an $n$-Venn diagram on the sphere that has antipodal symmetry; that is, the diagram is fixed by the map that takes a point on the sphere to the corresponding antipodal point. Thus, along with certain diagrams due to Anthony Edwards which can be drawn with rotational and reflective symmetry, for any isometry of the sphere that is an involution, there exists an $n$-Venn diagram on the sphere invariant under that involution. Our construction uses a recursively defined chain decomposition of the Boolean lattice.