  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.
