On maximal sublattices of finite lattices
The author investigates the relationships between finite distributive lattices, their maximal sublattices and the corresponding remainders. If \(L\) is a lattice and \(M\) a maximal sublattice of \(L\), then \(L\) is a minimal extension of \(M\), and the remainder \(R\) is defined as \(R= L\setminus M\). \(R\) is a sublattice of \(L\) whenever \(L\) is distributive. It is shown that a finite distributive lattice \(M\) has only finitely many minimal extensions \(L\) preserving 0 and 1. An algorithm is given to effectively list the minimal extensions. In contrast, given a finite distributive lattice \(R\), there are infinitely many pairs \((M,L)\) so that \(R\approx L\setminus M\). Again, a list of these pairs can be generated effectively. Birkhoff duality is the tool used to establish these results. It is known that \(|R|\leq|L|/3\). In this paper it is shown that equality occurs exactly if \(L\approx K\times 3\) and \(M\approx K\times 2\) for some finite distributive lattice \(K\). The author cites \textit{M. E. Adams, P. Dwinger} and \textit{J. Schmid}, ``Maximal sublattices of finite distributive lattices'' [Algebra Univers. 36, No. 4, 488-504 (1996; Zbl 0901.06009)] and \textit{M. E. Adams, R. Freese, J. B. Nation} and \textit{J. Schmid}, ``Maximal sublattices and Frattini sublattices of bounded lattices'' [J. Austral. Math. Soc., Ser. A 63, No. 1, 110-127 (1997; Zbl 0887.06005)] and the references given in these two papers as a port of entry to the literature on the subject. For general finite lattices, not necessarily distributive, questions about extensions and remainders have been addressed in the second of these papers. Some of the results of that paper are discussed at the end of the present paper and some open questions are discussed.
scientific article; zbMATH DE number 1321798