Word problems in sums of monoids

In the talk I will investigate the decidability of word problems for amalgams (sums) of monoids. Word problems for amalgams of monoids are equivalent to sums of congruence relations on words. They are also equivalent to unions of equational theories over signatures in which only unary function symbols appear. In particular I will show: - decidability for cases in which the intersection monoid is a group or group with zero, - undecidability for a case in which the intersection is a 3-elementn right-zero monoid, - undecidability for an amalgam of finite monoids. If time allows, I will also describe my work with Mikolaj Bojanczyk on amalgams of commutative monoids.