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Complete endomorphisms of the lattice of pseudovarieties of finite semigroups

Published online by Cambridge University Press:  17 April 2009

Norman R. Reilly  and
Shuhua Zhang
Norman R. Reilly
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby BC V5A 1S6, Canada
Shuhua Zhang
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby BC V5A 1S6, Canada
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The main result established that the mapping V → V ∩ W (V ∈ ℒ(F)) is a complete endomorphism of the lattice ℒ(F) of pseudovarieties of finite semigroups for certain particular pseudovarieties W, including the pseudovariety of bands.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 2 , April 1997 , pp. 207 - 218
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Almeida, J., ‘On finite simple semigroups’, Proc. Edinburgh Math. Soc. 34 (1991), 204–215.CrossRefGoogle Scholar
[2]Almeida, J., Finite semigroups and universal algebra (World Scientific, Singapore, 1994).Google Scholar
[3]Arbib, M. (ed.), The algebraic theory of machines, languages and semigroups (Academic Press, New York, 1968).Google Scholar
[4]Auinger, K., Hall, T.E., Reilly, N.R. and Zhang, S., ‘Congruences on the lattice of pseudovarieties of finite semigroups’, Internat J. Algebra Comput. (to appear).Google Scholar
[5]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Math. Surveys No. 7 (Amer. Math. Soc., Providence, RI, Vol. I, 1961; Vol. II, 1967).Google Scholar
[6]Eilenberg, S., Automata, languages and machines B (Academic Press, New York, 1976).Google Scholar
[7]Grätzer, G., General lattice theory (Birkhaüser, Basel, 1978).CrossRefGoogle Scholar
[8]Hall, T.E. and Jones, P.R., On the lattice of varieties of bands of groups, Pacific J. Math. 91 (1980), 327–337.Google Scholar
[9]Howie, J.M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
[10]Jones, P.R., ‘On the lattice of varieties of completely regular semigroups’, J. Austral. Math. Soc. 35 (1983), 227–235.Google Scholar
[11]Jones, P.R., ‘Mal'cev products of varieties of completely regular semigroups’, J. Austral. Math. Soc. 42 (1987), 227–246.CrossRefGoogle Scholar
[12]Pastijn, F., ‘The lattice of completely regular semigroup varieties’, J. Austral. Math. Soc. 49 (1990), 24–42.Google Scholar
[13]Pastijn, F., ‘Pseudovarieties of completely regular semigroups’, Semigroup Forum 42 (1991), 1–46.Google Scholar
[14]Pastijn, F. and Petrich, M., ‘The congruence lattice of a regular semigroup’, J. Pure Appl. Algebra 53 (1988), 93–123.CrossRefGoogle Scholar
[15]Pastijn, F. and Trotter, P.G., ‘Lattices of completely regular semigroup varieties’, Pacific J. Math. 119 (1985), 191–214.Google Scholar
[16]Petrich, M. and Reilly, N.R., ‘Semigroups generated by certain operators on varieties of completely regular semigroups’, Pacific J. Math. 132 (1988), 151–175.CrossRefGoogle Scholar
[17]Petrich, M. and Reilly, N.R., ‘Operators related to E-disjunctive and fundamental completely regular semigroups’, J. Algebra 134 (1990), 1–27.Google Scholar
[18]Petrich, M. and Reilly, N.R., ‘The modularity of the lattice of varieties of completely regular semigroups and related representations’, Glasgow Math. J. 32 (1990), 137–152.Google Scholar
[19]Pin, J.-E., Varieties of formal languages (North Oxford Academic Press, London, 1986).Google Scholar
[20]Polák, L., ‘On varieties of completely regular semigroups I’, Semigroup Forum 32 (1985), 97–123; II 36 (1987), 253–284; III 37 (1988), 1–30.Google Scholar
[21]Reilly, N.R., ‘Varieties of completely regular semigroups’, J. Austral. Math. Soc. 38 (1985), 372–393.Google Scholar
[22]Reilly, N.R., ‘Completely regular semigroups’, in Lattices, semigroups and universal algebra, (Almeida, J. et al. , Editors)(Plenum Press, New York, 1990), pp. 225–242.CrossRefGoogle Scholar
[23]Reilly, N.R., ‘Bounds on the variety generated by completely regular syntactic monoids from finite prefix codes’, Theoret. Comput. Sci. 134 (1994), 189–208.Google Scholar
[24]Reilly, N.R. and Zhang, S., ‘Congruence relations on the lattice of existence varieties of regular semigroups’, J. Algebra 178 (1995), 733–759.Google Scholar
[25]Reilly, N.R. and Zhang, S., ‘Operators on the lattice of pseudovarieties of finite semigroups’, Semigroup Forum (to appear).Google Scholar
[26]Reiterman, J.The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), 1–10.CrossRefGoogle Scholar
[27]Trotter, P.G., ‘Subdirect decompositions of the lattice of varieties of completely regular semigroups’, Bull. Austral. Math. Soc. 39 (1989), 343–351.CrossRefGoogle Scholar