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.

翻译：我们提出了高效的计算方案，用于解决自由带元素的相等性检验、乘法运算及最小代表元计算问题。带是满足恒等式$x^2 \approx x$的半群，而自由带$\operatorname{FB}(k)$是$k$元生成带簇中的自由对象。Radoszewski与Rytter开发了线性时间算法以检验两个词是否表示同一自由带元素。本文描述另一种解决该问题的线性时间算法。所提出的算法利用同步确定性换能器表示词，该表示适用于自由带中的高效乘法（复杂度与字母表大小呈二次关系）。该表示还提供了一种以二次复杂度寻找表示给定自由带元素的短字典序最小词的方法。