On Semitopological De Morgan Residuated Lattices

The class of De Morgan residuated lattices was introduced by L. C. Holdon (Kybernetika 54(3):443-475, 2018), recently, many mathematicians have studied the theory of ideals or filters in De Morgan residuated lattices and some of them investigated the properties of De Morgan residuated lattices endowed with a topology. In this paper, we introduce the notion of semitopological De Morgan residuated lattice, we present some examples and by considering the notion of upsets, for any element a of a De Morgan residuated lattice L, there is a topology τa on L and we show that L endowed with the topology τa is semitopological with respect to ∨, ∧ and ⊙, and right topological with respect to → . Moreover, in the general case of residuated lattices we prove that L endowed with the topology τa is semitopological with respect to ⊙ and right topological with respect to → . Finally, we obtain some of the topological aspects of this structure such as L endowed with the topology τa is a T0-space, but it is not a T1-space or Hausdorff space.