Way Below Relation is Auxiliary Relation

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Lrt $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.


$\ll$ is auxiliary relation

where $\ll$ denotes the way below relation.


By Way Below implies Preceding:

$\forall x, y \in S: x \ll y \implies x \preceq y$

By Preceding and Way Below implies Way Below:

$\forall x, y, z, u \in S: x \preceq y \ll z \preceq u \implies x \ll u$

By Join is Way Below if Operands are Way Below:

$\forall x, y, z \in S: x \ll z \land y \ll z \implies x \vee y \ll z$

By Bottom is Way Below Any Element:

$\forall x: \bot \ll x$

where $\bot$ denotes the smallest element in $L$.

Thus by definition:

$\ll$ is auxiliary relation.