## Word problems and a homological finiteness condition for monoids.(English)Zbl 0648.20045

A terminating Church-Rosser presentation (also known as a complete rewriting system) for a monoid S consists of a set R of ordered pairs of words in a free monoid F. This free monoid, together with the set of equalities arising from the pairs in R, should present S.
If u,v$$\in F$$, then v is a subword of u if $$u=avb$$ for some a,b$$\in F$$. A single replacement then consists of replacing a subword which is a left hand side of a pair in R by the right hand side of the pair. We require that there is no infinite sequence of such replacements. There is then a corresponding set of irreducible elements for which no replacement is possible and we further require that the natural map from irreducible elements of F to elements of S be bijective.
In this interesting paper, such rewriting systems are exploited to give the beginnings - up to dimension 3 - of a free $${\mathbb{Z}}S$$-resolution of $${\mathbb{Z}}$$. In the case of groups, this extends the well-known partial resolution which arises from the so-called ‘relation sequence’. Thus the term in dimension 0 is $${\mathbb{Z}}S$$ and in dimensions 1 and 2 is a free module on a basis which is bijective with, respectively, the generators of F, and R. The author adds a term in dimension 3 for which the basis is bijective with the set of all overlaps of left-hand sides of rules if R. He then defines a reasonably natural (although not unique) boundary map and shows the necessary exactness.
The result can also be used to give an example of group or monoid with a finite presentation and solvable word problem which can have no finite terminating Church-Rosser presentation. For the result shows that if S has a finite terminating Church-Rosser presentation then there must be a $${\mathbb{Z}}S$$-resolution of $${\mathbb{Z}}$$ which is finitely generated in the first 3 dimensions; i.e. S has type $$(FP)_ 3$$.
The interested reader should, however, be aware of results of D. J. Anick [Trans. Am. Math. Soc. 296, 641-659 (1986; Zbl 0598.16028)] which appear to be related. With somewhat different, but apparently related, hypotheses, Anick gives a $${\mathbb{Z}}S$$-resolution of $${\mathbb{Z}}$$ (in all dimensions). Where the hypotheses overlap, this resolution seems to agree with the one given here in the first three dimensions.
Reviewer: J.R.J.Groves

### MSC:

 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20J05 Homological methods in group theory 20M05 Free semigroups, generators and relations, word problems

Zbl 0598.16028
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