Polynomial time multiplication and normal forms in free bands

TL;DR

This paper introduces a new linear time algorithm for equality checking in free bands, utilizing transducer representations to enable efficient multiplication and minimal element computation.

Contribution

It presents an alternative linear time algorithm for free band equality checking using transducers, improving computational efficiency for multiplication and normal form calculation.

Findings

Linear time algorithm for free band equality checking

Efficient multiplication using transducer representations

Quadratic complexity for minimal element computation

Abstract

We present efficient computational solutions to the problems of checking equality, performing multiplication, and computing minimal representatives of elements of free bands. A band is any semigroup satisfying the identity $x ^ 2 \approx x$ and the free band $\operatorname{FB}(k)$ is the free object in the variety of $k$-generated bands. Radoszewski and Rytter developed a linear time algorithm for checking whether two words represent the same element of a free band. In this paper we describe an alternate linear time algorithm for checking the same problem. The algorithm we present utilises a representation of words as synchronous deterministic transducers that lend themselves to efficient (quadratic in the size of the alphabet) multiplication in the free band. This representation also provides a means of finding the short-lex least word representing a given free band element with quadratic complexity.