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：我们针对自由带元素中的相等性检验、乘法运算及最小代表元计算问题，提出了高效的计算解决方案。带是指满足恒等式$x^2 \approx x$的半群，而自由带$\operatorname{FB}(k)$是$k$生成元带簇中的自由对象。Radoszewski和Rytter曾开发出线性时间算法，用于检验两个词是否表示自由带中的同一元素。本文描述了另一种解决该问题的线性时间算法。该算法利用词作为同步确定性转换器的表示形式，使得自由带中的乘法运算（复杂度为字母表大小的平方）变得高效。该表示方法还提供了以二次复杂度找到表示给定自由带元素的最短词典序最小词的手段。