A monoid is a set with an associative operation and an identity element (so it's like a group, except that it need not have inverses).
Then (Mk)P is the class of decision problems solvable by an NP machine with the following acceptance mechanism. The ith computation path (under some lexicographic ordering) outputs an element mi of Mk. Then the machine accepts if and only if m1m2...ms is the identity (where s is the number of paths).
Defined by Hertrampf [Her97], who also showed the following (in the special case M is a group):
No class.