S-Glued sums of lattices

Christian Herrmann; Dale R. Worley

paper

S-Glued sums of lattices

arXiv:2409.10738

Abstract

For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let $L_{0}$ be a lattice with largest element $u_{0}$ , $L_{1}$ be a lattice disjoint from $L_{0}$ with smallest element $v_{1}$ , and $a \in L_{0}$ , $b \in L_{1}$ such that the intervals $[a, u_{0}]$ and $[v_{1}, b]$ are isomorphic. Then, after identifying those intervals you obtain $L_{0} \cup L_{1}$ , a lattice structure whose partial order is the transitive relation generated by the partial orders of $L_{0}$ and $L_{1}$ . It is modular if $L_{0}$ and $L_{1}$ are modular. Since in this construction the index set ${0, 1}$ is essentially a chain, this work presents a method -- termed S-glued -- whereby a general family $L_{x} (x \in S)$ of lattices can specify a lattice with the small-scale lattice structure determined by the $L_{x}$ and the large-scale structure determined by $S$ . A crucial application is representing finite-length modular lattices using projective geometries.

19 pages, 8 figures. German original by Christian Herrmann, English translation by Dale R. Worley. Added a footnote 1 to give more detail about Dilworth's work on the subject. Added translator's notes at end