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 andgeneralizations. International Journal of Pure andApplied Mathematics, 74(1):55–119, 2012.