Preceding is Top in Ordered Set of Auxiliary Relations
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Theorem
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $\map {\it Aux} L$ be the set of all auxiliary relations on $S$.
Let $P = \struct {\map {\it Aux} L, \precsim}$ be an ordered set
where $\precsim \mathop = \subseteq \restriction_{\map {\it Aux} L \times \map {\it Aux} L}$
Then
- $\preceq \mathop = \top_P$
where $\top_P$ denotes the greatest element in $P$.
Proof
By Preceding is Auxiliary Relation:
- $\preceq \mathop \in \map {\it Aux} L$
By definition:
- $\preceq$ is lower bound for $\O$ in $P$
We will prove that:
- $\forall R \in \map {\it Aux} L: R$ is lower bound for $\O \implies R \mathop \precsim \preceq$
Let $R \in \map {\it Aux} L$
By condition $(i)$ of definition of auxiliary relation:
- $R \mathop \subseteq \preceq$
Thus by definition of $\precsim$:
- $R \mathop \precsim \preceq$
$\Box$
By definition of infimum:
- $\preceq \mathop = \inf_P \O$
Thus by Infimum of Empty Set is Greatest Element:
- $\preceq \mathop = \top_P$
$\blacksquare$
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_4:3