The - and -Orders of the Thompson-Higman Monoid Mk, 1 and their Complexity

We study the monoid generalizationMk,1 of the Thompson-Higman groups, and we characterize the Rand the L-preorder ofMk,1. Although Mk,1 has only one non-zero J -class and k−1 non-zero D-classes, the Rand the L-preorder are complicated; in particular, <R is dense (even within an L-class), and <L is dense (even within an R-class). We study the computational complexity of the Rand the L-preorder. When inputs are given by words over a finite generating set of Mk,1, the Rand the L-preorder decision problems are in P. The main result of the paper is that over a “circuit-like” generating set, the R-preorder decision problem of Mk,1 is Π P 2 -complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Π 2 -complete, whereas the injectiveness problem is coNP-complete.