This could be a very interesting question:

Does every algebraic theory T with decidable word problem admit a coherent
presentation?

Here a presentation is "coherent" if it is such when T is considered as a
monoidal category.  See, for instance

@article{DBLP:journals/corr/CurienM17,
  author    = {Pierre{-}Louis Curien and
               Samuel Mimram},
  title     = {Coherent Presentations of Monoidal Categories},
  journal   = {CoRR},
  volume    = {abs/1705.03553},
  year      = {2017},
  url       = {http://arxiv.org/abs/1705.03553},
  timestamp = {Wed, 07 Jun 2017 14:42:32 +0200},
  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/corr/CurienM17},
  bibsource = {dblp computer science bibliography, http://dblp.org}
}

Cheers,
Andrew