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

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