Abstract
The Bernoulli measure on strings is used to define height functions for the dense
$\mathcal{ R}$
- and
$\mathcal{ L}$
-orders of the Thompson-Higman monoids M
k,1. The measure can also be used to characterize the
$\mathcal{ D}$
-relation of certain submonoids of M
k,1. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the
$\mathcal{ R}$
- and
$\mathcal{ L}$
-height of an element of M
k,1 is investigated.