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Coherent Monoids

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Victoria Gould
Victoria Gould
Affiliation:
University of YorkHeslington, York YO1 5DD, United Kingdom
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Abstract

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This paper is concerned with a new notion of coherency for monoids. A monoid S is right coherent if the first order theory of right S-sets is coherent; this is equivalent to the property that every finitely generated S-subset of every finitely presented right S-set is finitely presented. If every finitely generated right ideal of S is finitely presented we say that S is weakly right coherent. As for the corresponding situation for modules over a ring, we show that our notion of coherency is related to products of flat left S-sets, although there are some marked differences in behaviour from the case for rings. Further, we relate our work to ultraproducts of flat left S-sets and so to the question of axiomatisability of certain classes of left S-sets.

We show that a monoid S is weakly right coherent if and only if the right annihilator congruence of every element is finitely generated and the intersection of any two finitely generated right ideals is finitely generated. A similar result describes right coherent monoids. We use these descriptions to recognise several classes of (weakly) right coherent monoids. In particular we show that any free monoid is weakly right (and left) coherent and any free commutative monoid is right (and left) coherent.

MSC classification

Secondary: 20M10: General structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 2 , October 1992 , pp. 166 - 182
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bulman-Fleming, S. and McDowell, K., ‘Absolutely flat semigroups’, Pacific J. Math. 107 (1983), 319–333.CrossRefGoogle Scholar
[2]Bulman-Fleming, S. and McDowell, K., ‘Left absolutely flat generalized inverse semigroups’, Proc. Amer. Math. Soc. 94 (1985), 553–561.CrossRefGoogle Scholar
[3]Bulman-Fleming, S. and McDowell, K., ‘Monoids over which all weakly flat acts are flat’, Proc. Edinburgh Math. Soc. 33 (1990), 287–298.CrossRefGoogle Scholar
[4]Bulman-Fleming, S. and McDowell, K., ‘Coherent monoids’, in: Lattices, Semigroups and Universal Algebra, ed. Almeida, J. et al. (Plenum Press, N.Y. 1990).Google Scholar
[5]Chase, S. U., ‘Direct product of modules’, Trans. Amer. Math. Soc. 97 (1960), 457–473.CrossRefGoogle Scholar
[6]Chang, C. C. and Keisler, H. J., Model Theory (North-Holland, Amsterdam, New-York, Oxford 1973).Google Scholar
[7]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys 7 (American Mathematical Society, 1961).Google Scholar
[8]Faith, C., Rings, Modules and Categories I, (Springer-Verlag, Berlin, 1973).Google Scholar
[9]Fountain, J. B., ‘A class of right PP monoids’, Quart. J. Math. Oxford 28 (1977), 285–300.CrossRefGoogle Scholar
[10]Fountain, J. B., ‘Abundant Semigroups’, Proc. London Math. Soc. 44 (1982), 103–129.CrossRefGoogle Scholar
[11]Gould, V. A. R., ‘Axiomatisability problems for S-systems’, J. London Math. Soc. 35 (1987), 193–201.CrossRefGoogle Scholar
[12]Gould, V. A. R., ‘Model companions of S-systems’, Quart. J. Math. Oxford 38 (1987), 189–211.CrossRefGoogle Scholar
[13]Howie, J. M., An introduction to semigroup theory (Academic Press, 1976).Google Scholar
[14]Kilp, M., ‘Commutative monoids all of whose principal ideals are projective’, Semigroup Forum 6 (1973), 334–339.CrossRefGoogle Scholar
[15]Knauer, U., ‘Projectivity of acts and Morita equivalence of monoids’, Semigroup Forum 3 (1972), 359–370.CrossRefGoogle Scholar
[16]Knauer, U. and Petrich, M., ‘Characterization of monoids by torsion-free, flat, projective and free acts’, Archiv der Math. 36 (1981), 289–294.CrossRefGoogle Scholar
[17]Lambek, J., Lectures on Rings and Modules, (Blaisdell Publishing Co., 1976).Google Scholar
[18]Normak, P., ‘On noetherian and finitely presented polygons’, Uc. Zap. Tartu Gos. Univ. 431 (1977), 37–46.Google Scholar
[19]Redei, L., Theorie der endlich erzeugbaren kommutativen Halbgruppen, Hamburger mathematische Einzelschrifiten, Heft 41 (Physica-Verlag, Warzburg 1963).Google Scholar
[20]Rotman, J., An Introduction to Homological Algebra (Academic Press, New-York, 1979).Google Scholar
[21]Stenstrom, B., ‘Flatness and localisation over monoids’, Math. Nach. 48 (1971), 315–334.CrossRefGoogle Scholar
[22]Wheeler, W. H., ‘Model companions and definability in existentially complete structures’, Israel J. Math. 25 (1976), 305–330.CrossRefGoogle Scholar