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Nondualisable semigroups

Published online by Cambridge University Press:  17 April 2009

David Hobby
David Hobby
Affiliation:
Department of Mathematics, SUNY, New Paltz, NY 12561, United States of America, e-mail: hobbyd@newpaltz.edu
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Abstract

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An infinite family of finite semigroups is studied. It is shown that most of them do not generate a quasivariety which admits a natural duality.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 3 , June 2002 , pp. 491 - 502
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Berge, C., Graphs and hypergraphs (North-Holland, Amsterdam, London, 1973).Google Scholar
[2]Burris, S. and Sankappanavar, H.P., A course in universal algebra, Graduate Texts in Mathematics 78 (Springer-Verlag, New York, Heidelberg, Berlin, 1981).CrossRefGoogle Scholar
[3]Clark, D.M. and Davey, B.A., Natural dualities for the working algebraist (Cambridge University Press, Cambridge, 1998).Google Scholar
[4]Clark, D.M., Davey, B.A., and Pitkethly, J.G., ‘Binary homomorphisms and natural dualities’, J. Pure Appl. Algebra (to appear).Google Scholar
[5]Clark, D.M., Davey, B.A., and Pitkethly, J.G., ‘Dualisability of three-element unary algebras’, Internat. J. Algebra Comput. (to appear).Google Scholar
[6]Clark, D.M., Idziak, P., Sabourin, L., Szabó, Cs., and Willard, R., ‘Natural dualities for quasi-varieties generated by a finite commutative ring’, Algebra Universalis 46 (2001), 285–320.CrossRefGoogle Scholar
[7]Davey, B.A., ‘Dualisability in general and endodualisability in particular’, in Logic and Algebra (Pontignano, 1994), (Ursini, A. and Aglianò, P., Editors), Lecture Notes in Pure and Applied Mathematics 180 (Marcel Dekker, New York, 1996), pp. 437–455.Google Scholar
[8]Davey, B.A., Heindorf, L., and McKenzie, R., ‘Near unanimity: an obstacle to general duality theory’, Algebra Universalis 33 (1995), 428–439.CrossRefGoogle Scholar
[9]Davey, B.A. and Knox, B.J., ‘Regularising natural dualities’, Acta Math. Univ. Comenian. 68 (1999), 295–318.Google Scholar
[10]Davey, B.A. and Quackenbush, R.W., ‘Natural dualities for dihedral varieties’, J. Austral. Math. Soc. Ser. A 61 (1996), 216–228.CrossRefGoogle Scholar
[11]Davey, B.A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, in Contributions to lattice theory (Szeged, 1980) (North-Holland, Amsterdam, 1983), pp. 101–275.Google Scholar
[12]Davey, B.A. and Willard, R., ‘The dualisability of a quasi-variety is independent of the generating algebra’, Algebra Universalis 45 (2001), 103–106.CrossRefGoogle Scholar
[13]Köbler, J., Schöning, U. and Torán, J., The graph isomorphism roblem: Its structural complexity (Birkhäuser, Boston, 1993).CrossRefGoogle Scholar
[14]McKenzie, R.M., McNulty, G.F. and Taylor, W.F., Algebras, lattices, varieties Vol. 1 (Wadsworth and Brooks, Cole, Monterey, California, 1987).Google Scholar
[15]Pontryagin, L.S., ‘Sur les groupes abélian continus’, C.R. Acad. Sci. Paris 198 (1934), 238–240.Google Scholar
[16]Pontryagin, L.S., ‘The theory of topological commutative groups’, Ann. of Math. 35 (1934), 361–388.CrossRefGoogle Scholar
[17]Priestley, H.A., ‘Representation of distributive lattices by means of ordered Stone spaces’, Bull. London Math. Soc. 2 (1970), 186–190.CrossRefGoogle Scholar
[18]Priestley, H.A., ‘Ordered topological spaces and the representation of distributive lattices’, Proc. London Math. Soc. 3 24 (1972), 507–530.CrossRefGoogle Scholar
[19]Quackenbush, R. and Szabó, Cs., ‘Finite nilpotent groups are not dualizable’, (preprint, 1997).Google Scholar
[20]Saramago, M., A study of natural dualities, including an analysis of the structure of failsets, Ph.D. Thesis (Universidade de Lisboa, Portugal, 1998).Google Scholar
[21]Stone, M.H., ‘The theory of representations for Boolean algebras’, Trans. Amer. Math. Soc. 4 (1936), 37–111.Google Scholar