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Coherency, free inverse monoids and related free algebras†

Published online by Cambridge University Press:  09 September 2016

VICTORIA GOULD  and
MIKLÓS HARTMANN
VICTORIA GOULD
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mail: [email protected]
MIKLÓS HARTMANN
Affiliation:
Department of Mathematics, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. e-mail: [email protected]
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Abstract

A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.

Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.

Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.

The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 23 - 45
Copyright
Copyright © Cambridge Philosophical Society 2016 

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Footnotes

†

The authors acknowledge the support of EPSRC grant no. EP/I032312/1. Research also partially supported by the Hungarian Scientific Research Fund (OTKA) grant no. K83219.

References

REFERENCES

[1] Bulman-Fleming, S. and McDowell, K. Coherent monoids. In Lattices, Semigroups and Universal Algebra ed. Almeida, J. et al. (Plenum Press, N.Y., 1990), pp 29–37.Google Scholar
[2] Chase, S. U. Direct product of modules. Trans. Amer. Math. Soc. 97 (1960), 457–473.Google Scholar
[3] Choo, K. G., Lam, K. Y. and Luft, E. On free product of rings and the coherence property,. Algebraic K-theory, II:“Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math. 342 (Springer, Berlin, 1973), pp. 135–143.Google Scholar
[4] Cohn, P. M. Free rings and their relations. (Academic Press, London & New York, 1971).Google Scholar
[5] Cornock, C. Restriction Semigroups: Structure, Varieties and Presentations. PhD. thesis. University of York (2011).Google Scholar
[6] Eklof, P. and Sabbagh, G. Model-completions and modules. Annals of Math. Logic 2 (1971), 251–295.Google Scholar
[7] Fountain, J. B. Free right type A semigroups. Glasgow Math. J. 33 (1991), 135–148.Google Scholar
[8] Fountain, J. B., Gomes, G. M. S. and Gould, V. The free ample monoid. I.J.A.C. 19 (2009), 527–554.Google Scholar
[9] Gomes, G. M. S. and Gould, V. Graph expansions of unipotent monoids. Comm. Algebra 28 (2000), 447–463.CrossRefGoogle Scholar
[10] Gould, V. Model Companions of S-systems. Quart. J. Math. Oxford. 38 (1987), 189–211.Google Scholar
[11] Gould, V. Axiomatisability problems for S-systems. J. London Math. Soc. 35 (1987), 193–201.Google Scholar
[12] Gould, V. Coherent monoids. J. Australian Math. Soc. 53 (1992), 166–182.Google Scholar
[13] Gould, V., Hartmann, M. and Ruškuc, N. The free monoid is coherent. http://arxiv.org/abs/1412.7340 Proc. Edinburgh Math. Soc., to appear.Google Scholar
[14] Howie, J. M. Fundamentals of Semigroup Theory (Oxford University Press, 1995).Google Scholar
[15] Kilp, M., Knauer, U. and Mikhalev, A. V. Monoids, Acts and Categories. (de Gruyter, Berlin, 2000).Google Scholar
[16] Lawson, M. V. Inverse semigroups: The Theory of Partial Symmetries (World Scientific 1998).Google Scholar
[17] Munn, W. D. Free inverse semigroups. Proc. London Math. Soc. 29 (1974), 385–404.Google Scholar
[18] Scheiblich, H. E. Free inverse semigroups. Semigroup Forum 4 (1972), 351–358.Google Scholar
[19] Wheeler, W. H. Model companions and definability in existentially complete structures. Israel J. Math. 25 (1976), 305–330.Google Scholar