Same as AP, but for logarithmic-space instead of polynomial-time.
Defined in [GS90]. Equals mP by definition.
The definition of this class, due to [GS90], is not obvious. First, a monotone nondeterministic Turing machine is one such that, whenever it can make a transition with a 0 on its input tape, it can also make that same transition with a 1 on its input tape. (This restriction does not apply to the work tape.) A monotone alternating Turing machine is subject to the restriction that it can only reference an input bit x by, "there exists a z at most x," or "for all z at least x."
Then applying the result of [CKS81] that P = AL, mP is defined to be mAL: the class of decision problems solvable by a monotone alternating log-space Turing machine.
Actually there's a caveat: A monotone Turing machine or circuit can first negate the entire input, then perform a monotone computation. That way it becomes meaningful to talk about whether a monotone complexity class is closed under complement.
Strictly contained in mNP [Raz85].
Deciding whether a bipartite graph has a perfect matching, despite being both a monotone problem and in P, requires monotone circuits of superpolynomial size [Raz85b]. Letting MONO be the class of monotone problems, it follows that mP is strictly contained in MONO ∩ P.
See also: mNC1, mL, mNL, mcoNL.