8. The Influence of the Ordering

The ordering chosen for the computation of the examples presented in the previous section is a length-lexicographical one. But this is not necessarily the best one. First of all, it is only one of several different length-lexicographic orderings possible which was selected more or less randomly. Further, two other types of orderings, namely Knuth-Bendix orderings (kbo) and two kinds of syllable orderings which were presented in Section 5, are implemented in MRC 1.2. This allows us to study which influence the ordering has on the maximal/total number of cosets to be defined. Here, we neglect the fact, that these orderings are more complex and therefore use more time than the length-lexicographical one. As already mentioned time and space are not considered right now.

The examples were computed using different kbos, length-lexicographical and syllable orderings. For the length-lexicographical ordering and the syllable orderings at least one example was computed. The orderings together with the results are tabulated in Appendix C.

We use the following abbreviations: kbo-x means a Knuth-Bendix ordering where x has the greatest weight attached, normally all other weights are one. The abbreviation ll-ZzXxYy describes the length-lexicographical ordering with precedence while syl-l-ZzXxYy and syl-r-ZzXxYy describe the syllable orderings with the precedence being as for the length-lexicographical ordering and the l, respectively r, designating that the syllables are compared from left to right, respectively right to left.