The Frobenius Problem in a Free Monoid

TL;DR

This paper explores the Frobenius problem within free monoids, revealing that the bounds on the largest non-representable element can grow exponentially or subexponentially, contrasting with the quadratic bounds in the classical case.

Contribution

It extends the Frobenius problem to noncommutative free monoids and analyzes the growth behavior of the largest non-representable element.

Findings

Bound on g is exponential or subexponential depending on the measure.

Contrasts with quadratic bounds in the classical commutative case.

Abstract

The classical Frobenius problem is to compute the largest number g not representable as a non-negative integer linear combination of non-negative integers x_1, x_2, ..., x_k, where gcd(x_1, x_2, ..., x_k) = 1. In this paper we consider generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for an analogue of g, depending on the particular measure chosen.