On Semitopological De Morgan Residuated Lattices

The class of De Morgan residuated lattices was introduced by L‎. ‎C‎. ‎Holdon (Kybernetika ۵۴(۳):۴۴۳-۴۷۵‎, ‎۲۰۱۸)‎, ‎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 \tau_{a} on L‎ ‎and we show that L endowed with the topology‎ ‎\tau_{a} is semitopological with respect to \vee‎, ‎\wedge and \odot, and right topological with respect to \rightarrow.‎ ‎Moreover‎, ‎in the general case of residuated lattices we prove that L endowed with the topology‎ ‎\tau_{a} is semitopological with respect to \odot and right topological with respect to \rightarrow.‎ ‎Finally‎, ‎we obtain some of the topological‎ ‎aspects of this structure such as L endowed with the topology‎ ‎\tau_{a} is a \mathbf{T_۰}-space‎, ‎but it is not a \mathbf{T_۱}-space or Hausdorff space‎.

Keywords:

Residuated lattice‎ , ‎De Morgan laws‎ , ‎De Morgan residuated lattice‎ , ‎Filter‎ , ‎Semitopological algebras‎ , ‎Hausdorff space‎