On lattices embeddable into lattices of order-convex sets. II: Star-like posets
The authors continue the study of lattices that embed in lattices of order-convex subsets of posets. This paper is a continuation of Part I [Int. J. Algebra Comput. 17, No. 8, 1667--1712 (2007; Zbl 1152.06002)]. A poset is \textit{tree-like} if (i) whenever \(a, b \in P\) and \(a\) is majorized by \(b\) there is an ascending path from \(a\) to \(b\) in the Hasse diagram, and (ii) the Hasse diagram contains no circuits. The poset \(P\) is \textit{star-like} if it is tree-like and contains no subposet isomorphic to the four-element poset \(N\). Given a class \({\mathcal K}\) of posets, \({\mathbf{SUB}}({\mathcal K})\) denotes the class of sublattices of convexity lattices of posets in \({\mathcal K}\). With the class of star-like posets denoted by \({\mathcal S}\), the main result is that the finite lattices in \({\mathbf{SUB}}({\mathcal S})\) are characterized among all finite lattices by a finite collection of lattice identities. Such identities are given, and these include three identities that were previously shown to characterize the lattices embeddable in the lattice of order-convex sets of some poset, together with four other identities verified in the current paper. A similar characterization is determined for lattices embeddable in lattices of order-convex subsets of star-like lattices of length bounded by a postive integer \(m\).
scientific article; zbMATH DE number 5988078