Reading through Jean-Eric Pin's "Mathematical Foundations of Automata Theory". Love this book. However, I am confused by the following section, and am hoping for some clarity and more examples if possible. I need this for my research.

Syntactic order definition

I am not sure exactly how it is used to create the syntactic order in the following example:

Example

Putting the above example into this automaton:

• Q = {1,2,3}
• I = {1}
• F = {3}
• A = {a,b}
• E = {(1,a,2),(2,a,2),(2,b,3),(3,b,3),(3,a,2)}

For the above statements $u \le v $ and $s, t \in A*$,

$$ sut \in L \Rightarrow svt \in L $$

Would this look like the following from the example above?

$(1,a,2) \Rightarrow (1,aa,2)$

Does this define the relations from the example above? That is, are the relations from the example above (i.e. 'aa = a', etc.) syntactic congruence?

And, how are they used to define the syntactic order?

Reading through Jean-Eric Pin's "Mathematical Foundations of Automata Theory". Love this book. However, I am confused by the following section, and am hoping for some clarity and more examples if possible. I need this for my research.

Syntactic order definition

I am not sure exactly how it is used to create the syntactic order in the following example:

Example

Putting the above example into this automaton:

• $Q = \{1,2,3\}$
• $I = \{1\}$
• $F = \{3\}$
• $A = \{a,b\}$
• $E = \{(1,a,2),(2,a,2),(2,b,3),(3,b,3),(3,a,2)\}$

For the above statements $u \le v $ and $s, t \in A*$, $$ sut \in L \implies svt \in L $$ Would this look like the following from the example above?

$(1,a,2) \implies (1,aa,2)$

Does this define the relations from the example above? That is, are the relations from the example above (i.e. '$aa = a$', etc.) syntactic congruence?

And, how are they used to define the syntactic order?