Existence of finitely presented monoids with minimal relators and undecidable word problem

Construct a finitely presented monoid with two defining relations that has undecidable word problem, and determine whether there exists a finitely presented monoid with a single defining relation that has undecidable word problem.

Background

Undecidability results for monoids and groups are classically encoded via rewriting systems. The smallest number of relations necessary to force undecidability in monoids is unknown. These constructions relate to the minimal presentations yielding undecidable word problems in groups via embedding techniques.