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Expansions of inverse semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Mark V. Lawson ,
Stuart W. Margolis  and
Benjamin Steinberg
Mark V. Lawson
Affiliation:
Division of Mathematics, School of Informatics, University of Wales, Gwynedd LL57 1UT, Wales, e-mail: [email protected]
Stuart W. Margolis
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel, e-mail: [email protected]
Benjamin Steinberg
Affiliation:
Department of Pure Mathematics, Faculty of Science, University of Porto, 4099-002 Porto, Portugal
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Abstract

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We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

Keywords

Inverse semigroupsexpansionsF-inverse semigroupsprehomomorphismsBirget-Rhodes expansion.

MSC classification

Secondary: 20M18: Inverse semigroups 20M17: Regular semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 2 , April 2006 , pp. 205 - 228
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Billhardt, B., ‘On a wreath product embedding and idempotent pure congruences of inverse semigroups’, Semigroup Forum 45 (1992), 45–54.CrossRefGoogle Scholar
[2]Birget, J. C. and Rhodes, J., ‘Almost finite expansions of arbitrary semigroups’, J. Pure Appl. Algebra 32 (1984), 239–287.CrossRefGoogle Scholar
[3]Birget, J. C. and Rhodes, J., ‘Group theory via global semigroup theory’, J. Algebra 120 (1989), 284–300.CrossRefGoogle Scholar
[4]Blyth, T. S. and Janowitz, M. F., Residuation theory, International Series of Monographs in Pure and Applied Mathematics 102 (Pergamon Press, Oxford, 1972).Google Scholar
[5]Brown, T. C., ‘An interesting combinatorial method in the theory of locally finite semigroups’, Pacific J. Math. 36 (1971), 285–289.Google Scholar
[6]Evans, T., ‘Some connections between residual finiteness, finite embeddability and the word problem’, J. London Math. Soc. 1 (1969), 399–403.CrossRefGoogle Scholar
[7]Exel, R., ‘Partial actions of groups and actions of inverse semigroups’, Proc. Amer. Math. Soc. 126 (1998), 3481–3494.CrossRefGoogle Scholar
[8]Gomes, G. M. S. and Szendrei, M. B., ‘Idempotent pure extensions by inverse semigroups via quivers’, J. Pure Appl. Algebra 127 (1998), 15–38.CrossRefGoogle Scholar
[9]Kellendonk, J. and Lawson, M. V., ‘Partial actions of groups’, Internat. J. Algebra Comput. 14 (2004), 87–114.CrossRefGoogle Scholar
[10]Lawson, M. V., Inverse Semigroups: The theory of partial symmetries (World Scientific, Singapore, 1998).CrossRefGoogle Scholar
[11]Margolis, S. W. and Meakin, J. C., ‘E-unitary inverse monoids and the Cayley graph of a group presentation’, J. Pure Appl. Algebra 58 (1989), 45–76.Google Scholar
[12]Margolis, S. W. and Meakin, J. C., ‘Inverse monoids, trees, and context-free languages’, Trans. Amer. Math. Soc. 335 (1993), 259–276.Google Scholar
[13]Margolis, S. W., Meakin, J. C. and Stephen, J. B., ‘Free objects in certain varieties of inverse semigroups’, Canad. J. Math. 42 (1990), 1084–1097.CrossRefGoogle Scholar
[14]McFadden, R., ‘On homomorphisms of partially ordered semigroups’, Acta Sci. Math. Szeged 28 (1967), 241–249.Google Scholar
[15]O'Carroll, L., ‘A class of congruences on a posemigroup’, Semigroup Forum 3 (1971), 652–666.Google Scholar
[16]Petrich, M., Inverse semigroups (Wiley, New York, 1984).Google Scholar
[17]Petrich, M. and Reilly, N., ‘A network of congruences on an inverse semigroup’, Trans. Amer. Math. Soc. 270 (1982), 309–325.CrossRefGoogle Scholar
[18]Schein, B. M., ‘Completions, translational hulls, and ideal extensions of inverse semigroups’, Czechoslovak Math. J. 23 (1973), 575–610.CrossRefGoogle Scholar
[19]Steinberg, B., ‘Factorization theorems for morphisms of ordered groupoids and inverse semigroups’, Proc. Edinburgh Math. Soc. 44 (2001), 549–569.Google Scholar
[20]Steinberg, B., ‘Finite state automata: A geometric approach’, Trans. Amer. Math. Soc. 353 (2001), 3409–3464.CrossRefGoogle Scholar
[21]Stephen, J. B., ‘Presentations of inverse monoids’, J. Pure Appl. Algebra 63 (1990), 81–112.CrossRefGoogle Scholar
[22]Szendrei, M. B., ‘A note on Birget-Rhodes expansion of groups’, J. Pure Appl. Algebra 58 (1989), 93–99.CrossRefGoogle Scholar