Mapping Preserves Finite and Filtered Infima
Theorem
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be meet semilattices.
Let $f: S_1 \to S_2$ be a mapping.
Let $f$ preserve finite infima and preserve filtered infima.
Then $f$ also preserves all infima
Proof
Assume that
- $(1): \quad f$ preserves finite infima
and
- $(2): \quad f$ preserves filtered infima.
Let $X$ be a subset of $S_1$.
Let $X$ admits an infimum in $\left({S_1, \preceq_1}\right)$
Define $Z := \left\{ {\inf A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$
where
- $\inf A$ denotes the infimum of $A$ in $\left({S_1, \preceq_1}\right)$
- ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$
By Existence of Non-Empty Finite Infima in Meet Semilattice:
- for every non-empty finite subset $A$ of $X$, $A$ admits an infimum in $\left({S_1, \preceq_1}\right)$
- $\inf Z = \inf\bigcup \left({ {\it Fin}\left({X}\right) \setminus \left\{ {\varnothing}\right\} }\right) = \inf X$
- $Z$ admits an infimum in $\left({S_1, \preceq_1}\right)$
We will prove that
- $Z$ is filtered
Let $x, y \in Z$.
By definition of $Z$:
- $\exists A \in {\it Fin}\left({X}\right) \setminus \left\{ {\varnothing}\right\}: x = \inf A$
and
- $\exists B \in {\it Fin}\left({X}\right) \setminus \left\{ {\varnothing}\right\}: y = \inf B$
By Finite Union of Finite Sets is Finite:
- $A \cup B$ is finite
- $A \cup B \ne \varnothing$
By definition of $Z$:
- $\inf\left({A \cup B}\right) \in Z$
- $\left({\inf A}\right) \wedge \left({\inf B}\right) = \inf\left({A \cup B}\right)$
Thus by definition of infimum:
- $\inf\left({A \cup B}\right) \preceq_1 x \land \inf\left({A \cup B}\right) \preceq_1 y$
Thus by definition
- $Z$ is filtered
By $(2)$ and definitions of mapping preserves filtered infima and mapping preserves infimum on subset:
- $f^\to\left({Z}\right)$ admits an infimum in $\left({S_2, \preceq_2}\right)$ and $\inf\left({f^\to\left({Z}\right)}\right) = f\left({\inf Z}\right)$
We will prove that
- $X \subseteq Z$
Let $x \in X$.
By definition of ${\it Fin}\left({X}\right)$:
- $\left\{ {x}\right\} \in {\it Fin}\left({X}\right)$ and $\left\{ {x}\right\} \ne \varnothing$
By definition of $Z$:
- $\inf \left\{ {x}\right\} \in Z$
Thus by Infimum of Singleton:
- $x \in Z$
By Image of Subset under Relation is Subset of Image/Corollary 2:
- $f^\to\left({X}\right) \subseteq f^\to\left({Z}\right)$
By definition of infimum:
- $\inf\left({f^\to\left({Z}\right)}\right)$ is a lower bound for $f^\to\left({Z}\right)$
By Lower Bound is Lower Bound for Subset
- $\inf\left({f^\to\left({Z}\right)}\right)$ is a lower bound for $f^\to\left({X}\right)$
We will prove that
- for every lower bound $a$ for $f^\to\left({X}\right)$, $a$ is lower bound for $f^\to\left({Z}\right)$
Assume that
- $a$ is lower bound for $f^\to\left({X}\right)$
Let $x \in f^\to\left({Z}\right)$.
By definition of image of set:
- $\exists y \in S_1: y \in Z \land f\left({y}\right) = x$
By definition of $Z$:
- $\exists A \in {\it Fin}\left({X}\right): y = \inf A \land A \ne \varnothing$
By $(1)$ and definition of mapping preserves finite infima:
- $f$ preserves the infimum of $A$
By Existence of Non-Empty Finite Infima in Meet Semilattice
- $A$ admits an infimum in $\left({S_1, \preceq_1}\right)$ and $\inf\left({f^\to\left({A}\right)}\right) = f\left({\inf A}\right) = x$
By Image of Subset under Relation is Subset of Image/Corollary 2:
- $f^\to\left({A}\right) \subseteq f^\to\left({X}\right)$
By Lower Bound is Lower Bound for Subset
- $a$ is lower bound for $f^\to\left({A}\right)$
Thus by definition of infimum:
- $a \preceq_2 x$
Thus by definition
- $a$ is lower bound for $f^\to\left({Z}\right)$
By definition of infimum:
- for every lower bound $a$ for $f^\to\left({X}\right)$, $a \preceq_2 f\left({\inf X}\right)$
Again by definition of infimum:
- $f^\to\left({X}\right)$ admits an infimum in $\left({S_2, \preceq_2}\right)$
- $\inf\left({f^\to\left({X}\right)}\right) = f\left({\inf X}\right)$
Thus the result follows by definition of mapping preserves all infima.
$\blacksquare$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_0:71