Abstract
Squier [10] considered homotopy relations on the paths of the derivation graphs associated with monoid presentations in connection with the word problem. He introduced the notion of FDT (finite derivation type), which is a certain finiteness condition on homotopy of monoid presentations. He proved that FDT is an intrinsic property of a monoid not depending on its presentation. Moreover he proved that if a monoid admits a finite complete presentation, then it has FDT, and thus he could prove that his monoid S1 with solvable word problem has no finite complete presentation by showing that it does not have FDT.
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Kobayashi, Y. (2000). Homotopy Reduction Systems for Monoid Presentations II: The Guba—Sapir Reduction and Homotopy Modules. In: Birget, JC., Margolis, S., Meakin, J., Sapir, M. (eds) Algorithmic Problems in Groups and Semigroups. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1388-8_8
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DOI: https://doi.org/10.1007/978-1-4612-1388-8_8
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