All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1
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Theorem
Let $\struct {S, \preceq}$ be a complete lattice.
Let $\struct {T, \precsim}$ be an ordered set.
Let $g: S \to T$ be an all infima preserving mapping.
Then:
- $g$ is an increasing mapping.
Proof
Define a mapping $d: T \to S$:
- $\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$
Let $x, y \in S$ such that
- $x \preceq y$
By Upper Closure is Decreasing:
- $y^\succeq \subseteq x^\succeq$
By Infimum of Upper Closure of Element:
- $\map \inf {x^\succeq} = x$ and $\map \inf {y^\succeq} = y$
By definition of all infima preserving mapping:
- $g$ preserves the infimum on $x^\succeq$
and:
- $g$ preserves the infimum on $y^\succeq$
By definition of infimum on subset preserving mapping:
- $\map \inf {\map {g^\to} {x^\succeq} } = \map g x$ and $\map \inf {\map {g^\to} {y^\succeq} } = \map g y$
By Image of Subset under Mapping is Subset of Image:
- $\map {g^\to} {y^\succeq} \subseteq \map {g^\to} {x^\succeq}$
Thus by Infimum of Subset:
- $\map g x \precsim \map g y$
$\blacksquare$
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_1:14