Reflectional Topology in Residuated Lattices

Abstract

In this paper, we introduce soaker filters in a residuated lattice, give some characterizations and investigate some properties of them. Then we define a topology on the set of all the soaker filters, which we call reflectional topology, show it is an Alexandrov topology and give a basis for it. We introduce the notion of join-soaker filters and prove that when the residuated lattice is a join-soaker filter, then the reflectional topology is compact. We also give a characterization of connectedness of the reflectional topology. Finally, we prove the reflectional topology is [Formula: see text], give necessary and sufficient conditions under which it is [Formula: see text] and prove that being [Formula: see text] is equivalent to being [Formula: see text]. Several illustrative examples are given. </jats:p