Structure Induced by Lattice Operations is Lattice

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Theorem

Let $\struct {L, \vee, \wedge, \preceq}$ be a lattice.

Let $S$ be a set.

Let $\struct {L^S, \veebar, \barwedge}$ be the structure on $T^S$ induced by $\vee$ and $\wedge$.

Let $\precsim$ be the ordering on $S$ defined by:

$\forall f, g \in L^S: f \precsim g$ if and only if $f \veebar g = g$

as on Semilattice Induces Ordering.

Then $\struct {L^S, \veebar, \barwedge, \precsim}$ is a lattice.

Proof

$\struct {L^S, \veebar}$ and $\struct {L^S, \barwedge}$ are Semilattices

By definition of lattice:

$\struct {L, \vee}$ and $\struct {L, \wedge}$ are semilattices.

From Structure Induced by Semilattice Operation is Semilattice:

$\struct {L^S, \veebar}$ and $\struct {L^S, \veebar}$ are semilattices.

$\Box$

Absorption Laws

By definition of lattice:

$\vee$ absorbs $\wedge$

and:

$\wedge$ absorbs $\vee$

From Structure Induced by Absorbing Operations is Absorbing

$\veebar$ absorbs $\barwedge$

and:

$\barwedge$ absorbs $\veebar$

$\Box$

It follows that $\struct {L^S, \veebar, \barwedge, \precsim}$ is a lattice by definition.

$\blacksquare$