Topologies on residuated lattices

Abstract The main aim of this paper is to investigate the topologies that constructed by some ideals on residuated lattices and some topologies which induced by lattice ideals and distance functions on involutive residuated lattices. To begin with, we present that prime $\oplus $-ideals and prime $\boxplus $-ideals are coincident on $MTL$-algebras and give some new results about ideals on residuated lattices. In the following, we study an $i$-topology which is induced by an $i$-system on a residuated lattice $A$ and get that $A$ equipped with such an $i$-topology is a topological residuated lattice and give a characterization for such topological involutive residuated lattices. Meanwhile, we give a notion of $\mathcal {I}$-completion of a residuated lattice $A$ with respect to the $i$-topology induced by an $i$-system $\mathcal {I}$ and characterize the $\mathcal {I}$-completion of $A$ by means of the inverse limit of an inverse system. Finally, we show that the topology that induced by a lattice ideal and a distance function on an involutive residuated lattice is a semitopological residuated lattice and which coincides with some $i$-topological residuated lattice when the lattice ideal is an ideal of $A$.