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The Thompson-Higman groups G_{k,i} have a natural generalization to monoidsM_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features ofM_{k,i} and Inv_{k,i} and investigate the computational complexity of decisionproblems. The main interest of these monoids is their close connection withcircuits and circuit complexity.  The maximal subgroups of M_{k,1} are isomorphic to the groups G_{k,j} (1 \leqj \leq k-1); so we rediscover all the Thompson-Higman groups within M_{k,1}.  The Green relations \leq_J and \equiv_D of M_{k,1} can be decided indeterministic polynomial time when the inputs are words over a finitegenerating set of M_{k,1}.  When a circuit-like generating set is used for M_{k,1} then deciding \leq_Jis coDP-complete. The multiplier search problem for \leq_J isxNPsearch-complete, whereas the multiplier search problems of \leq_R and \leq_Lare not in xNPsearch unless NP = coNP.  Deciding \equiv_D for M_{k,1} when the inputs are words over a circuit-likegenerating set, is \oplus_{k-1}.NP-complete. For Inv_{k,1} over a circuit-likegenerating set, deciding \equiv_D is \oplus_{k-1} P-complete.

The Thompson-Higman monoids M_{k,i}: the J-order, the D-relation, and their complexity