Maximal submonoids of a
semigroup have recently attracted attention in semigroup literature. This is
particularly true for the semigroup ℬ(X) of binary relations on a set. The interesting
results of Zareckiǐ in this direction point to the fact that some of these statements
pertain to the more general situation of the translational hull of a Rees matrix
semigroup. More generally, we consider here maximal submonoids of the translational
hull of a regular semigroup.