Symbolic computation on the partition lattice of an \(n\)-set
Let \(n\) be a positive integer and \(S=\{1,\ldots , n\}\). A partition of \(S\) is a collection \(B=\{B_1, B_2, \ldots , B_k\}\) of non-empty disjoint subsets of \(S\) such that \(S=B_1\cup \cdots \cup B_k\). The subset \(B_i\) is called a block of the partition. The set of all partitions of \(S\) is denoted by \(\Pi_n\). One can make \(\Pi_n\) into a poset by defining \(A\prec B\) if every \(A_i\in A\) is contained in a \(B_j\in B\). The main purpose of this paper is to construct and study an injection from \(\Pi_n\) to a zero-dimensional affine algebra over \(\mathbb{F}_2\) as follows: Let \(I\) be a particular zero-dimensional ideal defined in the paper. Then, we define the injection map \[\Gamma_n:\Pi_n\longrightarrow \mathbb{F}_2[x_{i,j}\ | \ i=1,\ldots, n,j=1,\ldots ,n]/I\] \[A\mapsto [\prod_{i\text{ is smaller than }j\text{ and both lie in the same block}} x_{i,j}].\] Based on this presentation, interesting properties of this injection along with some properties of \(\Pi_n\) are revealed in an algebraic manner.
scientific article; zbMATH DE number 7074708