The following question was asked by me on the forum sci.math.research,

“An imprimitive group is a transitive permutation group with a non-trivial equivalence relation compatible with the action of the group.

Suppose we have a transitive transformation semigroup with a non-trivial equivalence relation compatible with the action of the transformation semigroup, do we call it an imprimitive semigroup? Or there is another name for it? “

Unfortunately, no one has responded this question over four weeks. Thus, I am trying MO to see if I have better luck here.

This ‘imprimitive’ property of some semigroups is one of the discoveries during my research on automata/semigroup theory. I have searched everywhere on Internet (including all the issues of Semigroup Forum I have access to) but could not find any mention of it. If anybody has seen this property before and can point to the source, I would appreciate it very much.