On quasiorder lattices and topology lattices of algebras

Abstract

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