Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.)

Let $A$ be its set of rank 1 elements (that is, the atoms, the elements covering the bottom element of the lattice). Let $a$ be in $A$ and let $B=A\setminus\{a\}$. Let $H$ be the set of rank $r-1$ elements of $L$ (that is, the co-atoms or hyperplanes, the elements covered by the top element of the lattice).

Assume that there is a 1-1 function $m$ from $B$ into $H$ such that for all $b$ in $B$, $b\le m(b)$. Further assume that if $n$ is a 1-1 function from $B$ into $H$ such that for all $b$ in $B$, $b\le n(b)$, then $n$ is onto. [That is, $(B,H,\le)$ is a critical society, in the language of Aharoni, Nash-Williams, and Shelah.]

Can you show that the above situation is impossible without using Bjorner's results (from his offprint, "Some Combinatorial Properties of Infinite Geometric Lattices")?


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