# Filtrator

A filtrator is a pair $(\mathfrak{A};\mathfrak{Z})$ consisting of a poset $\mathfrak{A}$ (the base of the filtrator) and its subset $\mathfrak{Z}$ (the core of the filtrator). The set $\mathfrak{Z}$ is considered as a poset with the induced order.

Having fixed a filtrator and an $a\in\mathfrak{A}$, we define:

 $\operatorname{up}a=\{X\in\mathfrak{Z}|X\geq a\}\quad\operatorname{down}a=\{X% \in\mathfrak{Z}|X\leq a\}.$

Probably the most important example of a filtrator is a primary filtrator that is the pair $(\mathfrak{F};\mathfrak{P})$ where $\mathfrak{F}$ is the set of filters on a poset ordered reverse to set-theoretic inclusion of filters and $\mathfrak{P}$ is the set of principal filters on this poset. For a filter $\mathcal{F}\in\mathfrak{F}$ we have $\operatorname{up}\mathcal{F}$ essentially equivalent with the set $\mathcal{F}$.

## References

• 1 Victor Porton. http://www.mathematics21.org/binaries/filters.pdfFilters on posets and generalizations. International Journal of Pure and Applied Mathematics, 74(1):55–119, 2012.
Title Filtrator Filtrator 2013-03-22 19:31:25 2013-03-22 19:31:25 porton (9363) porton (9363) 6 porton (9363) Definition msc 06B99 msc 06A06 msc 54A20 Filter Filter2 primary filtrator