Invariants ideals in Leavitt Path algebras

It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by $\text{Pl}(E)$, by $\text{Pc}(E)$ or by $\text{Pec}(E)$ are invariant under isomorphism. Though the ideal generated by $\text{Pb}(E)$ is not invariant we find its \lq\lq natural\rq\rq\ replacement (which is indeed invariant): the one generated by the vertices of $\text{Pbp}$ (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the $\text{tops}$ topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if $H$ is a hereditary saturated subset of vertices providing an invariant ideal, its exterior $\text{ext}(H)$ in the $\text{tops}$ topology of $E^0$ generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance $\text{Pl}$, etc). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain $\soc^{(1)}(
)\subset\soc^{(2)}(\ )\subset\cdots$ all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.