The kernel-trace approach to right congruences on an inverse semigroup

Petrich, Mario; Rankin, Stuart

doi:10.1090/S0002-9947-1992-1041051-6

The kernel-trace approach to right congruences on an inverse semigroup
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by Mario Petrich and Stuart Rankin PDF

Trans. Amer. Math. Soc. 330 (1992), 917-932 Request permission

Abstract:

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete $\cap$homomorphism but not a $\vee$-homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.

References

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR 0218472
Gérard Duchamp, Étude du treillis des congruences à droite sur le monoïde bicyclique, Semigroup Forum 33 (1986), no. 1, 31–46 (French). MR 834548, DOI 10.1007/BF02573180
J. B. Fountain and P. Lockley, Semilattices of groups with distributive congruence lattice, Semigroup Forum 14 (1977), no. 1, 81–91. MR 437660, DOI 10.1007/BF02194656
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
P. R. Jones, Semimodular inverse semigroups, J. London Math. Soc. (2) 17 (1978), no. 3, 446–456. MR 500629, DOI 10.1112/jlms/s2-17.3.446
P. R. Jones, Distributive inverse semigroups, J. London Math. Soc. (2) 17 (1978), no. 3, 457–466. MR 500630, DOI 10.1112/jlms/s2-17.3.457
John Meakin, One-sided congruences on inverse semigroups, Trans. Amer. Math. Soc. 206 (1975), 67–82. MR 369580, DOI 10.1090/S0002-9947-1975-0369580-9
William R. Nico, A classification of indecomposable $S$-sets, J. Algebra 54 (1978), no. 1, 260–272. MR 511464, DOI 10.1016/0021-8693(78)90029-7
Mario Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 752899

Right congruences on a Brandt semigroup

A classification of inverse semigroups whose full inverse subsemigroups form a chain

G. B. Preston, Inverse semi-groups, J. London Math. Soc. 29 (1954), 396–403. MR 64036, DOI 10.1112/jlms/s1-29.4.396
H. E. Scheiblich, Kernels of inverse semigroup homomorphisms, J. Austral. Math. Soc. 18 (1974), 289–292. MR 0360887
V. V. Vagner, The theory of generalized heaps and generalized groups, Mat. Sbornik N.S. 32(74) (1953), 545–632 (Russian). MR 0059267