Lattices of order-convex sets of forests
Let \((P,\leq)\) be a partially ordered set. A set \(A\subseteq P\) is order-convex if for any \(x,y\in A\), \(x \leq z\leq y\) implies that \(z\in A\). Let \(\mathbf{Co}(P)\) denote the lattice of convex subsets of \(P\) under inclusion; such lattices will be called convexity lattices. For a class \(\mathcal{K}\) of posets, let \(\mathbf{Co}(\mathcal {K})\) denote the class of lattices isomorphic to convexity lattices of posets from \(\mathcal {K}\). In ``Closure lattices of closure spaces'' [in: G.~Dorfer et al. (eds.), Proceedings of the 73rd workshop on general algebra ``73. Arbeitstagung Allgemeine Algebra'', 22nd conference of young algebraists, Alps-Adriatic-University of Klagenfurt, Austria, February 1--4, 2007. Klagenfurt: Verlag Johannes Heyn. Contributions to General Algebra 18, 175--188 (2008; Zbl 1148.06003)], the first author asks in particular about a characterization of the class of lattices belonging to \(\mathbf{Co}(\mathcal {K}\)) for certain classes of \(\mathcal {K}\) of posets. In this paper, the authors give such a characterization for \(\mathcal{K}\) being the class of forests, as well as for some other classes of posets. A poset \((P,\leq)\) is a forest if the lower set \(\downarrow x=\{ p\in P \mid p\leq x\}\) is a chain for any \(x\in P\). Equivalently, a poset \((P,\leq)\) is a forest if no two incomparable elements have a common upper bound in \((P,\leq)\). A connected forest is a tree. This paper provides a description of the class \(\mathbf{Co}\)(\(\mathcal {F}\)) (\(\mathbf{Co}(\mathcal{F}_{n}\)), \(n<\omega\), respectively) of lattices isomorphic to convexity lattices of posets which are forests (forests of length at most \(n\), respectively) as well as of the class \(\mathbf{Co}(\mathcal {L}\)) of lattices isomorphic to convexity lattices of linearly ordered posets. This characterization yields that the class of finite members from \(\mathbf{Co}(\mathcal{F}\)) (from \(\mathbf{Co}(\mathcal{F}_{n}\)), \(n<\omega\), or from \(\mathbf{Co}(\mathcal{L}\))) is finitely axiomatizable within the class of finite lattices.
scientific article; zbMATH DE number 5813768
scientific article