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FREE ADEQUATE SEMIGROUPS

Part of: Algebraic structures Semigroups

Published online by Cambridge University Press:  19 March 2012

MARK KAMBITES
MARK KAMBITES*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK (email: Mark.Kambites@manchester.ac.uk)
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Abstract

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We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

Keywords

adequate semigroupfree objectword problemMunn tree

MSC classification

Secondary: 20M10: General structure theory 08A50: Word problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 3 , December 2011 , pp. 365 - 390
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Branco, M. J. J., Gomes, G. M. S. and Gould, V. A. R., ‘Left adequate and left Ehresmann monoids’, Internat. J. Algebra Comput. 21(7) (2011), 1259–1284.CrossRefGoogle Scholar
[2]Cockett, J. R. B. and Guo, X., ‘Stable meet semilattice fibrations and free restriction categories’, Theory Appl. Categ. 16(15) (2006), 307–341 (electronic).Google Scholar
[3]Cockett, J. R. B. and Lack, S., ‘Restriction categories I. Categories of partial maps’, Theoret. Comput. Sci. 270(1–2) (2002), 223–259.CrossRefGoogle Scholar
[4]Cohn, P. M., Universal Algebra, 2nd edn, Mathematics and its Applications, Vol. 6 (D. Reidel, Dordrecht, 1981).CrossRefGoogle Scholar
[5]Foniok, J., Homomorphisms and structural properties of relational system. PhD Thesis, Faculty of Mathematics and Physics, Charles University, Prague, 2007.Google Scholar
[6]Fountain, J. B., ‘Adequate semigroups’, Proc. Edinb. Math. Soc. (2) 22(2) (1979), 113–125.Google Scholar
[7]Fountain, J. B., ‘Abundant semigroups’, Proc. Lond. Math. Soc. 44 (1982), 103–129.CrossRefGoogle Scholar
[8]Fountain, J. B., ‘Free right h-adequate semigroups’, in: Semigroups, Theory and Applications (Oberwolfach, 1986), Lecture Notes in Mathematics, Vol. 1320 (Springer, Berlin, 1988), pp. 97–120.Google Scholar
[9]Fountain, J. B., ‘Free right type A semigroups’, Glasgow Math. J. 33(2) (1991), 135–148.Google Scholar
[10]Fountain, J. B., Gomes, G. M. S. and Gould, V. A. R., ‘The free ample monoid’, Int. J. Algebra Comput. 19(4) (2009), 527–554.CrossRefGoogle Scholar
[11]Gomes, G. M. S. and Gould, V. A. R., ‘Left adequate and left Ehresmann monoids II’, J. Algebra 348(1) (2011), 171–195.CrossRefGoogle Scholar
[12]Gould, V. A. R., ‘(Weakly) left E-ample semigroups’. Notes available online at www-users.york.ac.uk/∼varg1/.Google Scholar
[13]Gutiérrez, C., ‘On free inverse semigroups’, Semigroup Forum 61(1) (2000), 154–158.CrossRefGoogle Scholar
[14]Harary, F. and Plummer, M. D., ‘On the core of a graph’, Proc. Lond. Math. Soc. (3) 17 (1967), 305–314.CrossRefGoogle Scholar
[15]Hell, P. and Nešetřil, J., ‘The core of a graph’, Discrete Math. 109(1–3) (1992), 117–126.CrossRefGoogle Scholar
[16]Kambites, M., ‘Retracts of trees and free left adequate semigroups’, Proc. Edinb. Math. Soc. 54 (2011), 731–747.Google Scholar
[17]Lawson, M. V., ‘Semigroups and ordered categories I: The reduced case’, J. Algebra 141 (1991), 422–462.Google Scholar
[18]Munn, W. D., ‘Free inverse semigroups’, Proc. Lond. Math. Soc. (3) 29 (1974), 385–404.Google Scholar
[19]Petrich, M., ‘Free inverse semigroups’, Colloq. Math. 59(2) (1990), 213–221.Google Scholar
[20]Poliakova, O. and Schein, B. M., ‘A new construction for free inverse semigroups’, J. Algebra 288(1) (2005), 20–58.Google Scholar
[21]Reynolds, M. A., ‘A new construction for free inverse semigroups’, Semigroup Forum 30(3) (1984), 291–296.Google Scholar
[22]Scheiblich, H. E., ‘Free inverse semigroups’, Semigroup Forum 4 (1972), 351–359.CrossRefGoogle Scholar
[23]Stallings, J., ‘Topology of a finite graph’, Invent. Math. 71 (1983), 551–565.Google Scholar