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

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

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

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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 $\cap$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.

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