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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

Citations:

Zbl 0598.16028
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References:

[1] Abels, H., An example of a finitely presented solvable group, (Wall, C. T.C., Homological Group Theory, London Mathematical Society Lecture Notes, 36, (1979), Cambridge University Press Cambridge), 205-211
[2] H. Abels and K.S. Brown, Finiteness properties of solvable S-arithmetic groups, Preprint. · Zbl 0617.20020
[3] Bieri, R., Homological dimension of discrete groups, Queen Mary College Mathematics Notes, (1976), London · Zbl 0357.20027
[4] Bieri, R., A connection between the integral homology and the centre of a rational linear group, Math. Z., 170, 263-266, (1980) · Zbl 0427.20042
[5] Book, R. V., Thue systems and the church-rosser property: replacement systems, specification of formal languages and presentations of monoids, (Cummings, L. J., Combinatorics on Words: Progress and Perspectives, (1983), Academic Press New York), 1-38 · Zbl 0563.68062
[6] Brown, K. S., Cohomology of groups, (1982), Springer Berlin · Zbl 0367.18012
[7] K.S. Brown, Finiteness properties of groups, Preprint. · Zbl 0613.20033
[8] Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton University Press Princeton, NJ · Zbl 0075.24305
[9] Church, A.; Rosser, J. B., Some properties of conversion, Trans. Amer. Math. Soc., 39, 472-482, (1936) · Zbl 0014.38504
[10] Cohn, P. M., Universal algebra, (1965), Harper and Row New York · Zbl 0141.01002
[11] Fox, R. H., Free differential calculus I, Ann. of Math., 57, 547-560, (1953) · Zbl 0142.22303
[12] Hilton, P. J.; Stammbach, U., A course in homological algebra, (1971), Springer Berlin · Zbl 0238.18006
[13] Houghton, C. H., The first cohomology of a group with permutation module coefficients, Arch. Math., 31, 254-258, (1978) · Zbl 0377.20044
[14] Huet, G., Confluent reductions: abstract properties and applications to term rewriting systems, J. Assoc. Comput. Mach., 27, 797-821, (1980) · Zbl 0458.68007
[15] Jantzen, M., A note on a special one-rule semi-thue system, Inform. Process. Lett., 21, 135-140, (1985) · Zbl 0581.68030
[16] Newman, M. H.A., On theories with a combinatorial definition of “equivalence”, Ann. of Math., 43, 223-243, (1943) · Zbl 0060.12501
[17] Nivat, M., Congruences parfaites et quasi-parfaites, Seminaire Dubreil, 7, (1971-1972) · Zbl 0338.02018
[18] Stallings, J. R., A finitely-presented group whose 3-dimensional integral homology is not finitely-generated, Amer. J. Math., 85, 541-543, (1963) · Zbl 0122.27301
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