Filter-induced linear topologies on residuated lattices: Hausdorffness, profiniteness, and finiteness conditions

Abstract

We study linear topologies on residuated lattices generated by systems of filters, with emphasis on the uniform structures and separation properties that they determine. A down-directed family of filters gives a natural compatible uniformity, and the associated topology makes the residuated lattice into a topological algebra. We characterize Hausdorffness by the triviality of the intersection of the underlying filter system. For compact topological residuated lattices, we prove the equivalence between topological profiniteness, residual finiteness, and representation as a closed subdirect product of finite discrete residuated lattices. We also analyze the descending chain condition ($DCC$) on filters. Under $DCC$, every filter system has a least element; hence every zero-dimensional linear topology is induced by a single filter, and the canonical map from filters to zero-dimensional linear topologies is bijective. This gives a corrected form of earlier representation arguments and identifies precisely where $DCC$ is required. Finally, working throughout in $\mathrm{ZFC}$, we give a sufficient criterion for the existence of non-discrete Hausdorff linear topologies, illustrated by the Gödel algebra.