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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A network of congruences on an inverse semigroup


Authors: Mario Petrich and Norman R. Reilly
Journal: Trans. Amer. Math. Soc. 270 (1982), 309-325
MSC: Primary 20M10
DOI: https://doi.org/10.1090/S0002-9947-1982-0642343-6
MathSciNet review: 642343
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Abstract: A congruence $ \rho $ on an inverse semigroup $ S$ is determined uniquely by its kernel and its trace. Denoting by $ {\rho ^{\min }}$ and $ {\rho _{\min }}$ the least congruence on $ S$ having the same kernel and the same trace as $ \rho $, respectively, and denoting by $ \omega $ the universal congruence on $ S$, we consider the sequence $ \omega $, $ {\omega ^{\min }}$, $ {\omega _{\min }}$, $ {({\omega ^{\min }})_{\min }}$, $ {({\omega _{\min }})^{\min }} \ldots $. These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on $ S$. We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.


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  • [1] D. G. Green, The lattice of congruences on an inverse semigroup, Pacific J. Math. 57 (1975), 141-152. MR 0390093 (52:10919)
  • [2] J. M. Howie, An introduction to semigroup theory, Academic Press, New York, 1976. MR 0466355 (57:6235)
  • [3] -, The maximum idempotent separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. 14 (1964), 71-79. MR 0163976 (29:1275)
  • [4] J. M. Howie and G. Lallement, Certain fundamental congruences on a regular semigroup, Proc. Glasgow Math. Assoc. 7 (1966), 145-156. MR 0197598 (33:5763)
  • [5] D. B. McAlister, E-unitary inverse semigroups over semilattices, Glasgow Math. J. 19 (1978), 1-12. MR 508341 (80b:20075)
  • [6] L. O'Carroll, Reduced inverse and partially ordered semigroups, J. London Math. Soc. 9 (1974), 293-301. MR 0360883 (50:13330)
  • [7] -, Strongly $ E$-reflexive inverse semigroups, Proc. Edinburgh Math. Soc. 20 (1976-77), 339-354. MR 0453899 (56:12152)
  • [8] M. Petrich, Congruences on inverse semigroups, J. Algebra 55 (1978), 231-256. MR 523456 (80d:20055)
  • [9] N. R. Reilly and W. D. Munn, $ E$-unitary congruences on inverse semigroups, Glasgow Math. J. 17 (1976), 57-75. MR 0404498 (53:8300)
  • [10] N. R. Reilly and H. E. Scheiblich, Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349-360. MR 0219646 (36:2725)
  • [11] H. E. Scheiblich, Kernels of inverse semigroup homomorphisms, J. Austral. Math. Soc. Ser. A 18 (1974), 289-292. MR 0360887 (50:13334)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0642343-6
Keywords: Inverse semigroups, congruences, lattice of congruences, implications, kernel and trace of a congruence
Article copyright: © Copyright 1982 American Mathematical Society

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