On equations and first-order theory of one-relator monoids

We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family F of one-relator monoids of the form 〈A|w=1〉 where for each monoid M in F, the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations) in M. We achieve this result by finding an interpretation in M of a free monoid, using only systems of equations together with length relations. It follows that each monoid in F has undecidable positive AE-theory, hence in particular it has undecidable first-order theory. The family F includes many one-relator monoids with torsion 〈A|w ⁿ=1〉 (n>1). In contrast, all one-relator groups with torsion are hyperbolic, and all hyperbolic groups are known to have decidable Diophantine problem. We further describe a different class of one-relator monoids with decidable Diophantine problem.