On modular homology in the Boolean algebra
Among various combinatorial principles leading to interesting structures and resulting formulas those based on the viewpoints derived from homological algebra are among the most productive and thus useful. In this paper the authors study rings \(R\) of characteristic \(p\) (not necessarily prime)\(>0\) as they interact with sets \(\Omega\) as a basis. The inclusion (boundary) map \(\partial: M_k\to M_{k-1}\) is given by \(\partial \Delta= \Gamma_1+ \dots+ \Gamma_k\), where the \(\Gamma_i\) are distinct subsets of \(\Delta\) of cardinal \((k-1)\). Properties of the resulting homology are greatly affected by the classification of \(\Omega\) as finite or infinite. In this paper the emphasis is on \(\Omega\) infinite and on prime characteristic \(p\). It is then shown that under these conditions \(0\leftarrow \dots \leftarrow 0\leftarrow M_0 \leftarrow\dots \leftarrow M_m\dots\) is \(p\)-exact for any \(m\). If \(S^\Omega\) denotes the full symmetric group on \(\Omega\) and if \(RS^\Omega\) denotes the resulting group-ring, then \(RS^\Omega\) acts naturally on the \(M_k\)'s and this action commutes with \(\partial\). If \(A\) is a subring of \(RS^\Omega\) and \(AM_k= \langle af\mid a\in A\), \(f\in M_k\rangle\), then \(\partial: A_k\to A_{k-1}\) as well, so that one obtains a sequence \(0\leftarrow A_0 \leftarrow A_1\leftarrow\dots \leftarrow A_k\dots\) . This allows one to specify ample rings \(A\) to be those for which \(0\leftarrow \dots\leftarrow 0\leftarrow A_0\leftarrow A_1\leftarrow \dots\leftarrow A_m\dots\) is a \(p\)-exact for all \(m\). One may also consider permutation groups \(G\) on \(\Omega\) for which an infinite subset \(\Omega^*\) exists such that \(G\) moves only a finite number of points from \(\Omega^*\) (finitary sections) and then use information obtained in applications leading to resulting structures and formulas such as one for the \(p\)-rank of the orbit inclusion matrix if \(G\) has finitely many orbits on \(k\)-element subsets. The paper is an important addition to the literature of its own subfield of investigation and an interesting example of finitary (but not necessarily finite!) combinatorics as it may be placed in a combinatorial setting whose infinite flavor is present and pleasant to some tastes.
scientific article; zbMATH DE number 840164
scientific article