Algorithmic Problems for Amalgams of Finite Semigroups

We prove that there exists an amalgam of two finite 4-nilpotent semigroups such that the corresponding amalgamated product has an undecidable word problem. We also show that the problem of embeddability of finite semigroup amalgams in any semigroups and the problem of embeddability of finite semigroup amalgams into finite semigroups are undecidable. We use several versions of Minsky algorithms and Slobodskojʹs result about undecidability of the universal theory of finite groups.