# Confusion in quotient structure on self-transition

Hi KS,

I have a question, I was going through the solution and was wondering why is there a self-loop/transition on S1S4 or s1={} (in your above diagram).

In the actual question, there is a transition from S1 to S4  but there is no transition from S4 to S1 or S1 to S1. So, I don't think that there should be a self-transition on S1S4.

As I understand for the final structure, there will be a transition from S1S4 to S2S5 if:

In the original question, there was a transition from:

1. Either S1 to S2 and S4 to S5

2. Or  S1 to S5 and S4 to S2

Am I wrong? Is this concept correct or not?

Thank you!
Looking forward to hearing from you
related to an answer for: Quotient Structure

+1 vote

We have computed the following equivalence relation:

```    B1 = {(S0,S0);(S1,S1);(S2,S2);(S3,S3);(S4,S4);(S5,S5);
(S0,S3);(S1,S4);(S2,S5);
(S3,S0);(S4,S1);(S5,S2)}```

That equivalence relation partitions the set of states into the following classes:

```    {S0;S3}
{S1;S4}
{S2;S5}
```

The quotient structure is a state transition system on these classes, and instead of using the above classes, we pick a representative of each class (which is the state with least index). The transitions between classes are added whenever the classes contain states which were connected by a transition.

The self-loop on S1 of the quotient structure is actually a self-loop from the class {S1;S4} to itself. Why is it there? Since we had a transition from S4 to S4 which is contained in the class of S1. Recall that the transitions of {S1;S4} (which is actually the state S1 in the quotient structure) are the transitions of S1 and also of S4.

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Hi KS,
Thank you for the quick reply, even on the weekend.

From What I understand from your explanation,

For example, for a set of states: S1, S2, S3, S4. In the merged states: S1S2 and S3S4.
if there was a transition from S1 to S3 in the original structure then, we will draw a transition from S1S2 to S3S4 in the final structure, whether there is a transition from S2 to S4 or not.
Did I understand it right?
Right! You got it!
Thank you very much :)