# Examples of Substructures that "do not know they are that substructure"

Just learned $$\mathbb{L}\vDash \mathbb{V}=\mathbb{L}$$ and was warned that this property is not obvious with the counterexample mentioned being $$HOD$$. I can think of a few examples of definable substructures that behave like $$\mathbb{L}$$. For example given a ring $$R$$ the center of that ring $$Z(R)$$ "knows" it's a center since $$Z(R)=Z(Z(R))$$. So we would have with much notational abuse $$Z(R)\vDash R=Z(R)$$. Similarly for a monoid $$M$$ if $$U(M)$$ is the set of all invertible elements then one would have that $$U(M)\vDash M=U(M)$$ since $$U(M)=U(U(M))$$. To avoid abuse of notation both examples are a case where I am given an $$L$$-structure $$M$$ satisfying some theory $$T$$ and there is an formula $$\varphi(x)$$ and the set $$\{x\in M: \varphi(x)\}$$ is a substructure satisfying $$T$$ which I will denote as $$\varphi(M)$$. What are some classical examples of $$\varphi(\varphi(M))\neq \varphi(M)$$ or $$\varphi(M)\not\vDash\forall x \varphi(x)$$ ?

We can call such proposition idempotent as suggested in the comments. It would also be nice to know what properties such sentence shared. For example "x is constructible" is $$\Delta_1$$

One I thought of right before posting were $$(\mathbb{Z},+)$$ and $$even(x)\equiv \exists y\;\; x=y+y$$ then $$even(\mathbb{Z})=2\mathbb{Z}$$ and $$even(even(\mathbb{Z}))=4\mathbb{Z}$$.

Edit: Also a suggestion for a better title would really be appreciated.

• hi aldo; just as a terminology suggestion, maybe you could call such formulas "idempotent"? (formulas where $\varphi(M)\models\forall x\varphi(x)$, I mean) May 12 at 23:46
• @AtticusStonestrom Yes I will add that May 13 at 0:07
• The set of non-isolated points in a topological space sort of counts (the issue being that some work is needed to make this appropriately first-order): the fact that this is not idempotent leads to the Cantor-Bendixson derivative. May 13 at 3:30