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PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

Part of: Semigroups

Published online by Cambridge University Press:  01 June 2009

VICTORIA GOULD  and
CHRISTOPHER HOLLINGS
VICTORIA GOULD
Affiliation:
Department of Mathematics, University of York, Heslington York YO10 5DD, UK (email: varg1@york.ac.uk)
CHRISTOPHER HOLLINGS*
Affiliation:
Centro de Álgebra da Universidade de Lisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: cdh500@cii.fc.ul.pt)
*
✉For correspondence; e-mail: cdh500@cii.fc.ul.pt
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Abstract

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We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

Keywords

augmented actionexpansionincomplete actioninverse semigrouppartial actionpremorphismweakly left E-ample semigroup

MSC classification

Secondary: 20M30: Representation of semigroups; actions of semigroups on sets

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 355 - 377
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Batbedat, A., ‘γ-demi-groupes, demi-modules, produit demi-direct’, in: Semigroups Proceedings: Oberwolfach 1979, Lecture Notes in Mathematics, 855 (Springer-Verlag, Berlin, 1981), pp. 1–18.Google Scholar
[2]Batbedat, A. and Fountain, J. B., ‘Connections between left adequate semigroups and γ-semigroups’, Semigroup Forum 22 (1981), 59–65.CrossRefGoogle Scholar
[3]Birget, J.-C. and Rhodes, J., ‘Almost finite expansions of arbitrary semigroups’, J. Pure Appl. Algebra 32 (1984), 239–287.Google Scholar
[4]Exel, R., ‘Partial actions of groups and actions of inverse semigroups’, Proc. Amer. Math. Soc. 126 (1998), 3481–3494.CrossRefGoogle Scholar
[5]Fountain, J., ‘A class of right PP monoids’, Quart. J. Math. Oxford (2) 28 (1977), 285–300.CrossRefGoogle Scholar
[6]Fountain, J., ‘Adequate semigroups’, Proc. Edinb. Math. Soc. (2) 22 (1979), 113–125.CrossRefGoogle Scholar
[7]Fountain, J. and Gomes, G. M. S., ‘The Szendrei expansion of a semigroup’, Mathematika 37 (1990), 251–260.CrossRefGoogle Scholar
[8]Fountain, J., Gomes, G. M. S. and Gould, V. A. R., ‘A Munn type representation for a class of E-semiadequate semigroups’, J. Algebra 218 (1999), 693–714.CrossRefGoogle Scholar
[9]Gilbert, N. D., ‘Actions and expansions of ordered groupoids’, J. Pure Appl. Algebra 198 (2005), 175–195.CrossRefGoogle Scholar
[10]Gomes, G. M. S., ‘The generalised prefix expansion of a weakly left ample semigroup’, Semigroup Forum 72 (2006), 387–403.CrossRefGoogle Scholar
[11]Gomes, G. M. S. and Gould, V., ‘Finite proper covers in a class of finite semigroups with commuting idempotents’, Semigroup Forum 66 (2003), 433–454.CrossRefGoogle Scholar
[12]Gould, V., ‘(Weakly) left E-ample semigroups’, http://www-users.york.ac.uk/∼varg1/finitela.ps.Google Scholar
[13]Hollings, C., ‘Partial actions of monoids’, Semigroup Forum 75 (2007), 293–316.Google Scholar
[14]Hollings, C., ‘Partial actions of semigroups and monoids’, PhD Thesis, University of York, 2007.CrossRefGoogle Scholar
[15]Jackson, M. and Stokes, T., ‘An invitation to C-semigroups’, Semigroup Forum 62 (2001), 279–310.Google Scholar
[16]Jackson, M. and Stokes, T., ‘Algebras of partial maps’, Proceedings of the Special Interest Meeting on Semigroups and Related Mathematics, University of Sydney, 2005, to appear.Google Scholar
[17]Kellendonk, J. and Lawson, M. V., ‘Partial actions of groups’, Internat. J. Algebra Comput. 14 (2004), 87–114.CrossRefGoogle Scholar
[18]Lawson, M. V., ‘Semigroups and ordered categories I: the reduced case’, J. Algebra 141 (1991), 422–462.Google Scholar
[19]Lawson, M. V., Margolis, S. W. and Steinberg, B., ‘Expansions of inverse semigroups’, J. Aust. Math. Soc. 80 (2006), 205–228.CrossRefGoogle Scholar
[20]Manes, E., ‘Guarded and banded semigroups’, Semigroup Forum 72 (2006), 94–120.CrossRefGoogle Scholar
[21]McAlister, D. B. and Reilly, N. R., ‘E-unitary covers for inverse semigroups’, Pacific J. Math. 68 (1977), 161–174.CrossRefGoogle Scholar
[22]Megrelishvili, M. and Schröder, L., ‘Globalisation of confluent partial actions on topological and metric spaces’, Topology Appl. 145 (2004), 119–145.Google Scholar
[23]Renshaw, J., ‘Inverse semigroups acting on graphs’, in: Semigroups and Languages (World Scientific, River Edge, NJ, 2004), pp. 212–239.Google Scholar
[24]Schein, B. M., ‘Restrictively multiplicative algebras of transformations’, Izv. Vysš. Učebn. Zaved. Mat. 4(95) (1970), 91–102 (in Russian).Google Scholar
[25]Schein, B. M., ‘Relation algebras and function semigroups’, Semigroup Forum 1 (1970), 1–62.Google Scholar
[26]Schweizer, B. and Sklar, A., ‘The algebra of functions’, Math. Ann. 139 (1960), 366–382.CrossRefGoogle Scholar
[27]Schweizer, B. and Sklar, A., ‘The algebra of functions II’, Math. Ann. 143 (1961), 440–447.CrossRefGoogle Scholar
[28]Schweizer, B. and Sklar, A., ‘The algebra of functions III’, Math. Ann. 161 (1965), 171–196.CrossRefGoogle Scholar
[29]Schweizer, B. and Sklar, A., ‘Function systems’, Math. Ann. 172 (1967), 1–16.CrossRefGoogle Scholar
[30]Szendrei, M. B., ‘A note on Birget–Rhodes expansion of groups’, J. Pure Appl. Algebra 58 (1989), 93–99.Google Scholar
[31]Trokhimenko, V. S., ‘Menger’s function systems’, Izv. Vysš. Učebn. Zaved. Mat. 11(138) (1973), 71–78 (in Russian).Google Scholar