  We describe and count the maximal subsemigroups of many well-known monoids of
transformations and monoids of partitions. More precisely, we find the maximal
subsemigroups of the full spectrum of monoids of order- or
orientation-preserving transformations and partial permutations considered by
V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer,
Jones, and Motzkin monoids; and certain further monoids.
  Although descriptions of the maximal subsemigroups of some of the
aforementioned classes of monoids appear in the literature, we present a
unified framework for determining these maximal subsemigroups. This approach is
based on a specialised version of an algorithm for determining the maximal
subsemigroups of any finite semigroup, developed by the third and fourth
authors. This allows us to concisely present the descriptions of the maximal
subsemigroups, and to more clearly see their common features.
