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# 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. Filters on posets and generalizations. International Journal of Pure and Applied Mathematics, 74(1):55–119, 2012.

Defines:

primary filtrator

Keywords:

order theory,partially ordered set

Related:

Filter,Filter2

Type of Math Object:

Definition

Major Section:

Research

## Mathematics Subject Classification

06B99*no label found*06A06

*no label found*54A20

*no label found*

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