Brouwerian Lattice iff Shift Mapping is Lower Adjoint
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Theorem
Let $\struct {S, \preceq}$ be a lattice.
Then $\struct {S, \preceq}$ is a Brouwerian lattice if and only if:
- $\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a lower adjoint
Proof
- $\struct {S, \preceq}$ is a Brouwerian lattice
- $\forall x, y \in S: x$ has relative pseudocomplement with respect to $y$ by definition of Brouwerian lattice
- $\forall x, y \in S: \max \set {s \in S: x \wedge s \preceq y}$ exists by definition of relative pseudocomplement
- $\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a lower adjoint by Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist
$\blacksquare$
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:def 19