Bernoulli measure on strings, and Thompson-Higman monoids

TL;DR

This paper explores the Bernoulli measure on strings to define height functions in Thompson-Higman monoids, analyzes their properties, and investigates the computational complexity of measure-related problems within these algebraic structures.

Contribution

It introduces a novel application of Bernoulli measure to Thompson-Higman monoids, characterizes the D-relation, and studies the complexity of measure computations.

Findings

Bernoulli measure effectively characterizes the D-relation in certain submonoids.

Computing the Bernoulli measure of specific sets is computationally complex.

Determining the R- and L-height of elements in M_{k,1} involves significant computational challenges.

Abstract

The Bernoulli measure on strings is used to define height functions for the dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can also be used to characterize the 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 R- and L-height of an element of M_{k,1} is investigated.