On the Ershov upper semilattice $\mathfrak{L}_E$

We find some links between $\Sigma$-reducibility and $T$-reducibility. We prove that (1) if a quasirigid model is strongly $\Sigma$-definable in a hereditarily finite admissible set over a locally constructivizable $B$-system, then it is constructivizable; (2) every abelian $p$-group and every Ershov algebra is locally constructivizable; (3) if an antisymmetric connected model is $\Sigma$-definable in a hereditarily finite admissible set over a countable Ershov algebra then it is constructivizable.