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.



A Nomenclature Issue : Imprimitive Semigroup?
The name is not commonly used, but there is no reason why you shouldn't use it since it is as good a name as any. The term primitive transformation monoid has been used to mean no nontrivial congruences. You might also want make clear what you mean by transitive since there are two possible ways to generalize the group theoretic concept to monoids, corresponding in automata theoretic terms to connected versus strongly connected. I would guess you mean the latter.


