The Subword Reversing Method

We summarize the main known results involving subword reversing, a method of semigroup theory for constructing van Kampen diagrams by referring to a preferred direction. In good cases, the method provides a powerful tool for investigating presented (semi)groups. In particular, it leads to cancellativity and embeddability criteria for monoids and to efficient solutions for the word problem of monoids and groups of fractions. Subword reversing is a combinatorial method for investigating presented semigroup. It has been developed in various contexts and the results are scattered in different sources [13, 15, 24, 17, 20, 25, 5, ...]. This text is a survey that discusses the main aspects of the method, its range, its uses, and its efficiency. The emphasis is put on the exportable applications rather than on the internal technicalities, for which we refer to literature. New examples and open questions are mentioned, as well as a few new results. Excepted in the cases where no reference is available, proofs are sketched, or just omitted. General context and main results. As is well known, working with a semigroup or a group presentation is usually very difficult, and most problems are undecidable in the general case. Subword reversing is one of the few methods that can be used to investigate a presented semigroup, possibly a presented group. The specificity of the method is that, in order to solve the word problem of a presented semigroup, or, equivalently, construct a van Kampen diagram for a pair of initially given words, one directly compares the words one to the other instead of separately reducing each of them to some normal form, as in standard approaches like Knuth–Bendix algorithm or Gröbner–Shirshov bases (see Figure 1). Every semigroup presentation is in principle eligible for subword reversing, but the method leads to useful results only when some condition called completeness is satisfied. The good news is that the completeness condition is satisfied in a number of nontrivial cases and that, even if it is not initially satisfied, it can be satisfied once a certain completion procedure has been performed. The general philosophy is that, whenever the completeness condition is fulfilled, some properties of the considered semigroup can be read from the presentation easily. Typically, when a presentation is complete, it is sufficient that the presentation contains no obvious obstruction to left-cancellativity, namely no relation of the form sv = sv with v 6= v, to be sure that the presented semigroup does admit left-cancellation. Combined with a completeness criterion (several exist), this leads to practical, easy to use, cancellativity criteria, such as the following one. Theorem 1 (a criterion for left-cancellatibility). Assume that a semigroup (or a monoid) M admits a presentation (S,R) satisfying the following conditions: Work partially supported by the ANR grant ANR-08-BLAN-0269-02 1991 Mathematics Subject Classification. 20B30, 20F55, 20F36.