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$E$-INVERSIVE SEMIGROUP
In this paper we investigate some subclasses of strongly regular congruences on an 
$E$-inversive semigroup 
$S$. We describe the minimum and the maximum strongly orthodox congruences on 
$S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on 
$S$ with characteristic trace 
$\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an 
$E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence.
