On quasiorder lattices and topology lattices of algebras

In this paper, it is shown that the dual $\widetilde{\mathrm{Qord}}\,\mathfrak A$ of the quasiorder lattice of any algebra $\mathfrak A$ is isomorphic to a sublattice of the topology lattice $\Im(\mathfrak A)$. Further, if $\mathfrak A$ is a finite algebra, then $\widetilde{\mathrm{Qord}}\,\mathfrak A\cong\Im(\mathfrak A)$. We give a sufficient condition for the lattices $\widetilde{\mathrm{Con}}\,\mathfrak A$, $\widetilde{\mathrm{Qord}}\,\mathfrak A$, and $\Im(\mathfrak A)$ to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.