On incomplete lattice homomorphisms in subspaces of geometries: "half" a problem of Hartmanis from 1959

Turing Award winner Juris Hartmanis introduced in 1959 lattices of subspaces of generalized partitions (“partitions of type n”; “geometries” if $n=2$). Hartmanis states it is “an unsolved problem whether there are any incomplete lattice homomorphisms in” lattices of subspaces of geometries. (He continues, “[I]f so how can these geometries be characterized.”) We give a positive answer to this question.