nLab
reverse lattice of filters

It is convenient for the poset $𝔉(A)$ of filters on a given set $A$ to have the reverse order $⊑$ to set-theoretic inclusion:

$$a⊑b\iff a\supseteq b.$$a \sqsubseteq b \Leftrightarrow a \supseteq b .

We will call it the reverse poset of filters (or reverse lattice when it is a lattice).

We can denote the lattice operations on $𝔉(A)$ as $⊑$, $\sqcap$, $\bigsqcup$, $⨅$, $⨆$. By $0^{𝔉(A)}$ and $1^{𝔉(A)}$ we denote the minimal and maximal elements of this poset. So set-theoretically, $𝒳\sqcap 𝒴$ is the filter generated by the union $X\cup Y$, $𝒳\bigsqcup 𝒴$ is the intersection $X\cap Y$, and similarly for ${⨅}_i{𝒳}_i$ and ${⨆}_i{𝒳}_i$, while $0^{𝔉(A)}$ is the power set $𝒫(A)$ and $1^{𝔉(A)}$ is the empty set $\varnothing$.