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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.

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up vote 2 down vote accepted

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.

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Yes, I do mean strongly connected. –  scaaahu Oct 4 '11 at 6:40
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