Recursively unsolvable word problems of modular lattices and diagram-chasing

Abstract

The unsolvability of the word problem for modular lattices is demonstrated, using the known unsolvability of the word problem for semigroups. More generally, suppose that Λ is a nontrivial ring, N is a denumerably infinite set, and Γ(ΛN) is the lattice of submodules of the left Λ-module of all functions N → Λ. Then a lattice presentation with a finite number of generators and relations is constructed that has a recursively unsolvable word problem in any quasivariety of modular lattices that contains Γ(ΛN).Given a finite commutative abelian category diagram with specified exactness conditions, it may be that other exactness conditions are implied. For example, consider the five-lemma. It is proved that no algorithm can compute every exactness implication in every finite commutative diagram with specified exactness conditions. The result is obtained essentially by expressing an unsolvable modular lattice word problem as a family of diagram-chasing problems.