Geodesic
The kernel relation for an extension of completely 0-simple semigroups
Glasgow mathematical journal, Tome 41 (1999) no. 2, pp. 211-230

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Let S be an (ideal) extension of a completely 0-simple semigroup S0 by a completely 0-simple semigroup S1. Congruences on S can be uniquely represented in terms of congruences on S0 and S1. In this representation, for a congruence ρ on S, we express ρK,ρT,ρK and ρT, where these denote the least (greatest) congruences with the same kernel (trace) as ρ. Let κ be the least completely 0-simple congruence on S. We provide necessary and sufficient conditions, in terms of the kernel of κ, in order that the relation K be a congruence, and also that [Cscr](S)/K be a modular lattice, where [Cscr](S) denotes the congruence lattice of S. 
Petrich, Mario. The kernel relation for an extension of completely 0-simple semigroups. Glasgow mathematical journal, Tome 41 (1999) no. 2, pp. 211-230. doi: 10.1017/S0017089599970866
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     title = {The kernel relation for an extension of completely 0-simple semigroups},
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     year = {1999},
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     doi = {10.1017/S0017089599970866},
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