ultrafilter

既定 英語（イギリス） (en-GB)

名詞

集合

A device that performs ultrafiltration. / (mathematics, of a set, whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements.

日本語の意味

超濾過装置：超濾過（液体中の微細な固体粒子の除去）を行うための装置。 / 極大フィルター：数学において、部分集合の包含順序が定義され、補集合に関して二分法則を持つ適切なフィルター。

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We installed a new ultrafilter to improve the plant's water purification efficiency.

工場の水処理効率を改善するため、新しい限外濾過装置を設置しました。

詳細

意味（1）

A device that performs ultrafiltration.

意味（2）

(mathematics, of a set, whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements.

A proper filter of a set has these three properties: (1) it does not contain the empty set, (2) if it contains a subset then it contains all supersets of that subset, and (3) if it contains a pair of subsets then it also contains their intersection. To make it into an ultrafilter it must be made as large as possible without including the empty set. That can be prevented by not allowing any pair of disjoint sets to be both included. If, given a pair of complementary subsets, one of them is prevented from being included, then all subsets of it should be prevented from being included as well, by the second rule. That takes care of all subsets disjoint from the other complementary subset, which should then be included, in order to make the filter approach maximality, i.e., turn it into an ultrafilter.

詳細

意味（3）

(mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.

An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset.

詳細

ultrafilters （ plural ）