Bottom Relation is Auxiliary Relation

Theorem

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

Let $B = \left\{ {\left({\bot, x}\right): x \in S}\right\}$

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

Then

$B$ is auxiliary relation.

Proof

By definition of smallest element:

$\forall x \in S: \bot \preceq x$

Thus by definition of $B$:

$\forall x, y \in S: \left({x, y}\right) \in B \implies x \preceq y$

We will prove that

$\forall x, y, z, u \in S: x \preceq y \land \left({y, z}\right) \in B \land z \preceq u \implies \left({x, u}\right) \in B$

Let $x, y, z, u \in S$ such that

$x \preceq y \land \left({y, z}\right) \in B \land z \preceq u$

By definition of $B$:

$y = \bot$

By definition of smallest element:

$\bot \preceq x$

By definition of antisymmetry:

$x = \bot$

Thus by definition of $B$:

$\left({x, u}\right) \in B$

$\Box$

We will prove that

$\forall x, y, z \in S: \left({x, z}\right) \in B \land \left({y, z}\right) \in B \implies \left({x \vee y, z}\right) \in B$

Let $x, y, z \in S$ such that

$\left({x, z}\right) \in B \land \left({y, z}\right) \in B$

By definition of $B$:

$x = y = \bot$

By Join is Idempotent:

$x \vee y = \bot$

Thus by definition of $B$:

$\left({x \vee y, z}\right) \in B$

$\Box$

Thus by definition of $B$:

$\forall x \in S: \left({\bot, x}\right) \in B$

Thus by definition:

$B$ is auxiliary relation.

$\blacksquare$

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:funcreg 7