nLab
reverse lattice of filters

It is convenient for the poset 𝔉(A)\mathfrak{F}(A) of filters on a given set AA to have the reverse order \sqsubseteq to set-theoretic inclusion:

abab. a \sqsubseteq b \Leftrightarrow a \supseteq b .

We will call it the reverse lattice of filters (since it is a lattice); compare reverse poset.

We can denote the lattice operations on 𝔉(A)\mathfrak{F}(A) as \sqsubseteq, \sqcap, \sqcup, ⨅, \bigsqcup. By 0 𝔉(A)0^{\mathfrak{F}(A)} and 1 𝔉(A)1^{\mathfrak{F}(A)} we denote the minimal and maximal elements of this poset. So set-theoretically, 𝒳𝒴\mathcal{X} \sqcap \mathcal{Y} is the filter generated by the union XYX \cup Y, 𝒳𝒴\mathcal{X} \sqcup \mathcal{Y} is the intersection XYX \cap Y, and similarly for i𝒳 i⨅_i \mathcal{X}_i and i𝒳 i\bigsqcup_i \mathcal{X}_i, while 0 𝔉(A)0^{\mathfrak{F}(A)} is the improper filter (the power set) 𝒫(A)\mathcal{P}(A) and 1 𝔉(A)1^{\mathfrak{F}(A)} is the filter generated by the empty set, which is {A}\{A\}.

Revised on September 13, 2013 19:50:33 by Toby Bartels (98.23.131.69)