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},
timestamp = {Wed, 07 Jun 2017 14:42:32 +0200},
}
Cheers,
Andrew