Filtrator
A filtrator is a pair $(\U0001d504;\u2128)$ consisting of a poset $\U0001d504$ (the base of the filtrator) and its subset $\u2128$ (the core of the filtrator). The set $\u2128$ is considered as a poset with the induced order.
Having fixed a filtrator and an $a\in \U0001d504$, we define:
$$\mathrm{up}a=\{X\in \u2128|X\ge a\}\mathit{\hspace{1em}}\mathrm{down}a=\{X\in \u2128|X\le a\}.$$ |
Probably the most important example of a filtrator is a primary filtrator that is the pair $(\U0001d509;\U0001d513)$ where $\U0001d509$ is the set of filters on a poset ordered reverse to set-theoretic inclusion of filters and $\U0001d513$ is the set of principal filters^{} on this poset. For a filter $\mathcal{F}\in \U0001d509$ we have $\mathrm{up}\mathcal{F}$ 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 |