Segment of Auxiliary Relation is Subset of Lower Closure

Theorem

Let $\struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $R$ be auxiliary relation on $S$.

Let $x \in S$.

Then

$x^R \subseteq x^\preceq$

where

$x^R$ denotes the $R$-segment of $x$,

$x^\preceq$ denotes the lower closure of $x$.

Proof

Let $a \in x^R$.

By definition of $R$-segment of $x$:

$\tuple {a, x} \in R$

By definition of auxiliary relation:

$a \preceq x$

Thus by definition of lower closure of element:

$a \in x^\preceq$

$\blacksquare$

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_4:12