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On existence varieties of locally inverse semigroups

Published online by Cambridge University Press:  24 October 2008

K. Auinger ,
J. Doyle  and
P. R. Jones
K. Auinger
Affiliation:
Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria
J. Doyle
Affiliation:
Mathematics Department, Monash University, Clayton 3168, Victoria, Australia
P. R. Jones
Affiliation:
Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53233, U.S.A.
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Abstract

A locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 197 - 217
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Auinger, K.. The bifree locally inverse semigroup on a set. J. Algebra (to appear).Google Scholar
[2]Auinger, K.. On the lattice of existence-varieties of locally inverse semigroups. Can. Math. Bull. (to appear).Google Scholar
[3]Auinger, K.. A system of bi-identities for locally inverse semigroups. (Preprint.)Google Scholar
[4]Clifford, A. H. and Preston, G. B.. The algebraic theory of semigroups. I. (Amer. Math. Soc., Providence, RI (1964).)Google Scholar
[5]Doyle, J.. On existence varieties of orthodox semigroups. J. Austral. Math. Soc. (to appear).Google Scholar
[6]Hall, T. E.. Congruences and Green's relations on regular semigroups. Glasgow Math. J. 13 (1972), 167–175.CrossRefGoogle Scholar
[7]Hall, T. E.. On regular semigroups. J. Algebra 24 (1972), 1–24.CrossRefGoogle Scholar
[8]Hall, T. E.. Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc. 40 (1989), 59–77.CrossRefGoogle Scholar
[9]Hall, T. E.. Regular semigroups: amalgamation and the lattice of existence varieties. Algebra Universalis 29 (1991), 79–108.CrossRefGoogle Scholar
[10]Hall, T. E.. A concept of variety for regular semigroups. In Semigroup Theory, Proceedings of the Monash University Conference on Semigroup Theory in Honor of G. P. Preston (eds. Hall, T. E., Jones, P. R. and Meakin, J. C.) (World Scientific, Singapore, 1991), pp. 101–116.Google Scholar
[11]Howie, J. M.. An Introduction to Semigroup Theory (Academic Press, London, 1976).Google Scholar
[12]Jones, P. R. and Trotter, P. G.. Joins of semigroup varieties and band varieties. Int. J. Algebra Comput. 1 (1991), 371–385.CrossRefGoogle Scholar
[13]Kaďourek, J. and Szendrei, M. B.. A new approach in the theory of orthodox semigroups. Semigroup Forum 40 (1990), 257–296.CrossRefGoogle Scholar
[14]McAlister, D. B.. Rees matrix covers for locally inverse semigroups. Trans. Amer. Math. Soc. 277 (1983), 727–738.CrossRefGoogle Scholar
[15]Meakin, J. C.. The free local semilattice on a set. J. Pure Appl. Algebra 27 (1983), 263–275.CrossRefGoogle Scholar
[16]Meakin, J. C.. Perfect rectangular bands of inverse monoids. J. Pure Appl. Algebra 40 (1986), 149–162.CrossRefGoogle Scholar
[17]Meakin, J. C.. Regular expansions of completely simple semigroups. J. Pure Appl. Algebra 40 (1986), 163–175.CrossRefGoogle Scholar
[18]Meakin, J. C.. The free rectangular band of inverse semigroups on a set. J. Pure Appl. Algebra 40 (1986), 259–273.CrossRefGoogle Scholar
[19]Meakin, J. C. and Pastijn, F.. The structure of pseudosemilattices. Algebra Universalis 13 (1981), 355–372.CrossRefGoogle Scholar
[20]Meakin, J. C. and Pastijn, F.. The free pseudosemilattice on two generators. Algebra Universalis 14 (1982), 297–309.CrossRefGoogle Scholar
[21]Mitsch, H.. A natural partial order for semigroups. Proc. Amer. Math. Soc. 97 (1986), 384–388.CrossRefGoogle Scholar
[22]Nambooripad, K. S. S.. Structure of regular semigroups. (Mem. Amer. Math. Soc. 224 (1979).)Google Scholar
[23]Nambooripad, K. S. S.. The natural partial order on a regular semigroup. Proc. Edinburgh Math. Soc. 23 (1980), 249–260.CrossRefGoogle Scholar
[24]Nambooripad, K. S. S.. Pseudosemilattices and biordered sets. I. Simon Stevin 55 (1981), 103–110.Google Scholar
[25]Nambooripad, K. S. S.. Pseudosemilattices and biordered sets. II. Simon Stevin 56 (1982), 143–160.Google Scholar
[26]Nambooripad, K. S. S.. Pseudosemilattices and biordered sets. III. Simon Stevin 56 (1982), 239–256.Google Scholar
[27]Pastijn, F.. Rectangular bands of inverse semigroups. Simon Stevin 56 (1982), 1–97.Google Scholar
[28]Pastijn, F.. The structure of pseudo-inverse semigroups. Trans. Amer. Math. Soc. 273 (1982), 631–655.CrossRefGoogle Scholar
[29]Pastijn, F. and Petrich, M.. Regular Semigroups as Extensions. (Pitman, London, 1985.)Google Scholar
[30]Pastijn, F. and Petrich, M.. Congruences on regular semigroups. Trans. Amer. Math. Soc. 295 (1986), 607–633.CrossRefGoogle Scholar
[31]Pastijn, F. and Petrich, M.. The congruence lattice of a regular semigroup. J. Pure Appl. Algebra 53 (1988), 93–123.CrossRefGoogle Scholar
[32]Petrich, M.. Inverse Semigroups (Wiley, New York, 1984).Google Scholar
[33]Petrich, M. and Reilly, N. R.. All varieties of central completely simple semigroups. Trans. Amer. Math. Soc. 280 (1983), 623–636.CrossRefGoogle Scholar
[34]Schein, B. M.. Pseudosemilattices and pseudolattices (Russian). Izv. Vysš. Učebn. Zaved. Matematika 117 (1972), 81–94;Google Scholar
Schein, B. M.. English transl. In Amer. Math. Soc. Transl. 119 (1983), 1–16.Google Scholar
[35]Trotter, P. G.. Congruence extensions in regular semigroups. J. Algebra 137 (1991), 166–179.CrossRefGoogle Scholar
[36]Yamada, M.. Regular semigroups whose idempotents satisfy permutation identities. Pacific J. Math. 21 (1967), 371–392,CrossRefGoogle Scholar
[37]Yamada, M.. On a certain class of regular semigroups (with a supplement of T. E. Hall), Proc. Symp. Regular Semigroups, Northern Illinois University (1979), 146–179.Google Scholar
[38]Yeh, Y. T.. The existence of e-free objects in e-varieties of regular semigroups. Int. J. Algebra Comput. 2 (1992), 471–484.CrossRefGoogle Scholar
[39]Yeh, Y. T.. On existence varieties of E-solid or locally inverse semigroups and e-invariant congruences. J. Algebra (to appear).Google Scholar
[40]Zhang, S.. Completely regular semigroup varieties of the form 〈U∘G〉 (Preprint).Google Scholar