THE ${\mathcal R}$- AND ${\mathcal L}$-ORDERS OF THE THOMPSON–HIGMAN MONOID Mk, 1 AND THEIR COMPLEXITY

Abstract

We study the monoid generalization Mk, 1 of the Thompson–Higman groups, and we characterize the [Formula: see text]- and the [Formula: see text]-order of Mk, 1. Although Mk, 1 has only one nonzero [Formula: see text]-class and k-1 nonzero [Formula: see text]-classes, the [Formula: see text]- and the [Formula: see text]-order are complicated; in particular, [Formula: see text] is dense (even within an [Formula: see text]-class), and [Formula: see text] is dense (even within an [Formula: see text]-class). We study the computational complexity of the [Formula: see text]- and the [Formula: see text]-order. When inputs are given by words over a finite generating set of Mk, 1, the [Formula: see text]- and the [Formula: see text]-order decision problems are in P. However, over a "circuit-like" generating set the [Formula: see text]-order decision problem of Mk, 1 is [Formula: see text]-complete, whereas the [Formula: see text]-order decision problem is coNP-complete. Similarly, for acyclic circuits the surjectiveness problem is [Formula: see text]-complete, whereas the injectiveness problem is coNP-complete.