S-Glued sums of lattices

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.