reverse lattice of filters

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

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

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

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

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