Algebra Seminar talk
2022-06-03
Josef Šlapal
On bases of closure operators on complete lattices
Abstract:
We will discuss closure operators on complete lattices and define for
them the concepts of closed and open bases as well as that of a neighborhood
base. Some properties of these concepts will be studied including relationships
between them. In particular, we will give sufficient conditions under which the
properties are analogous to those known for Kuratowski closure operators, i.e.,
topologies.
In the literature, many topological concepts and results can be found extended to a categorical level, i.e., to categories with closure operators. While neighborhoods and neighborhood bases with respect to a categorical closure operator were introduced and studied in a paper by E. Giuli and J. Slapal, the concepts of closed and open bases have not yet been considered for such an operator. These concepts, which will be introduced and studied for closure operators on complete lattices in the talk, may naturally be transferred to categorical closure operators because they are simply families of closure operators on the subobject lattices of the objects of the category considered.
All the results presented for closed, open, and neighborhood bases of closure operators on complete lattices may be aplied to categorical closure operators as well (though the subobject lattices may be large, becoming classes).