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

TL;DR

This paper addresses Hartmanis's 1959 problem by demonstrating the existence of incomplete lattice homomorphisms in lattices of subspaces of geometries, providing a significant advancement in understanding these structures.

Contribution

The paper proves that incomplete lattice homomorphisms do exist in lattices of subspaces of geometries, resolving a long-standing open problem posed by Hartmanis.

Findings

Existence of incomplete lattice homomorphisms confirmed

Characterization of geometries with such homomorphisms provided

Advances understanding of lattice structures in geometries

Abstract

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.