Extending L-Topologies to Bipolar L-Fuzzy Topologies

We develop a model that allows for the extension of an L-topology tau on a set X (i.e., tau subset of L-X) to a bipolar L-fuzzy topology T on this set (i.e., T:L-X -> L). This model is based on the use of an additional algebraic structure on a complete infinitely distributive lattice L, and the derived lattice L obtained by "bipolarizing" the original lattice L. The properties of the obtained bipolar L-fuzzy topology are studied. A number of examples show how the choice of algebraic structure on L affects the resulting bipolar L-fuzzy topology. In particular, we consider the case when the original lattice L is enriched with a structure of a Girard monoid. In this case our construction becomes most transparent. In addition, the relationship between the extended bipolar L-fuzzy topology and the corresponding extended bipolar L-fuzzy co-topology in this case becomes dual. In the last section we examine the proposed model from the categorical point of view.