On the word problem for tensor products and amalgams of monoids

We prove that the word problem for the free product with amalgamation S *(U) T of monoids can be undecidable, even when S and T are finitely presented monoids with word problems that are decidable in linear time, the factorization problems for U in each of S and T, as well as other problems, are decidable in polynomial time, and U is a free finitely generated unitary submonoid of both S and T. This is proved by showing that the equality problem for the tensor product Sx(U)T is undecidable and using known connections between tensor products and amalgams. We obtain similar results for semigroups, and by passing to semigroup rings, we obtain similar results for rings as well. The proof shows how to simulate an arbitrary Turing machine as a communicating pair of two deterministic pushdown automata and is of independent interest. A similar idea is used in a paper by E. Each to show undecidability of the tensor equality problem for modules over commutative rings.