
18 Partitioning filter objects 53
19 Open problems 54
19.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
19.2 Quasidifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
19.3 Non-formal problems . . . . . . . . . . . . . . . . . . . . . . . . . 56
Appendix A Some counter-examples 56
Appendix A.1 Weak and strong partition . . . . . . . . . . . . . . 57
Appendix B Logic of Generalizations 59
Appendix B.1 The formalistic . . . . . . . . . . . . . . . . . . . . 59
1. Preface
This article is intended to collect in one document the known properties of
filters on posets (and so me generalizations thereof, namely “filtrators” defined
below).
It seems that until now were published no reference on the theory of filters.
This text is to fill the gap.
This text will also serve as the reference base for my further articles. This
text provides a definitive place to refer as to the collection of theorems about
filters.
Detailed study of filters is required for my ongoing research w hich will be
published as ”Algebraic General Topology” series.
In place of studying filters in this article are instea d researched what the au-
thor calls “filter objects”. Filter objects a re basically the lattice of filters orde red
reverse to set inclusion, with principal filters equated with the poset element
which generates them. (See be low for formal definition of “filter objects”.)
Although our primary interest are properties of filters on a set, in this work
are instead re searched the mor e gene ral theory of “filtra tors” (see below).
This article also contains some original research:
• filtrators ;
• straight maps and separation subsets;
• other minor results, such as the theory of free stars.
2. Notation and basic results
We denote PS the set of all subsets of a set S.
hfi X
def
= {f x | x ∈ X} for any set X and function f.
3