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
Canonical name	Filtrator
Date of creation	2013-03-22 19:31:25
Last modified on	2013-03-22 19:31:25
Owner	porton (9363)
Last modified by	porton (9363)
Numerical id	6
Author	porton (9363)
Entry type	Definition
Classification	msc 06B99
Classification	msc 06A06
Classification	msc 54A20
Related topic	Filter
Related topic	Filter2
Defines	primary filtrator