Standard contexts and arrow-closed subcontexts of integer partition lattices.

This dataset is support material for the paper entitled "Lattices of integer partitions are almost subdirectly indecomposable". It provides standard contexts $\mathbb{K}$ of integer partition lattices $\mathcal{L}_n$ for $n\in \{7, \ldots, 42\}$ and as of $n=13$ for each of them their largest one-generated arrow-closed subcontext, which we termed the monster. The contexts are provided in Burmeister's ".cxt" format. Additionally, we provide some one-generated arrow-closed subcontexts in ".cex" format (saved by the Concept Explorer), which contain an arranged concept lattice. The Concept Explorer software is freely available and, according to the note FCA and the Concept Explorer in 2024, it can be installed and used on modern operating systems. These links are also listed in "related works" below. Background: This data set belongs to the area of Formal Concept Analysis, see also (Ganter and Wille, 1999). The standard contexts for integer partition lattices were defined in (Ganter, 2020), and others. So called arrow relations and arrow-closed subcontexts are defined in and after Definition 25 of (Ganter and Wille, 1999), respectively. How the data was generated: The .cxt files were generated by a custom (currently not included) Python code that generates standard contexts by defining all join- and meet irreducible integer partitions (originally characterised by (Brylawski, 1973), but also shown in (Almazaydeh et al., 2024, Lemma 2.11). The incidence relation used in the contexts is the dominance order, originally established by (Brylawski, 1973), also shown in (Almazaydeh et al., 2024, Definition 2.7). This gives the standard contexts. Given the standard context, the code defines all arrow relations which are characterised by the results in shown in (Almazaydeh et al., 2024, Table 3) and applies an arrow-closing algorithm find an arrow-closed subcontext.