On the word problem for tensor products and amalgams of monoids

Jean Camille Birget, Stuart W. Margolis, John Meakin

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

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 SØU 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. Bach to show undecidability of the tensor equality problem for modules over commutative rings.

Original languageEnglish
Pages (from-to)271-294
Number of pages24
JournalInternational Journal of Algebra and Computation
Volume9
Issue number3-4
DOIs
StatePublished - 1999

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