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

Authors:
Mario Petrich and Stuart Rankin

Journal:
Trans. Amer. Math. Soc. **330** (1992), 917-932

MSC:
Primary 20M18

DOI:
https://doi.org/10.1090/S0002-9947-1992-1041051-6

MathSciNet review:
1041051

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Abstract: A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete homomorphism but not a -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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1041051-6

Keywords:
Right congruences,
inverse semigroup,
regular semigroup,
kernel,
trace,
right kernel system,
Clifford semigroup

Article copyright:
© Copyright 1992
American Mathematical Society