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