  It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by
$\Pl(E)$, by $\Pc(E)$ or by $\Pec(E)$ are invariant under isomorphism. Though
the ideal generated by $\Pb(E)$ is not invariant we find its \lq\lq
natural\rq\rq\ replacement (which is indeed invariant): the one generated by
the vertices of $\Pbp$ (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 $\tops$
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 $\ext(H)$ in the $\tops$ 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 $\Pl$, 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.
