Every lattice induces a partial ordering

up:: Lattice

Given a lattice $(X,\wedge ,\vee )$, one can always create a Partial Ordering $\le$ as

$$x\le y⟺x=x\wedge y⟺y=x\vee y$$

It can be read as a subset ordering: $X\subseteq Y$ if $X$ itself is its intersection with $Y$, and also if $Y$ is itself the union with $X$ (which is already within it).

Proof

Given $\le$, prove that it is a partial ordering:

• Reflexive: $\forall x\in X:x=x\wedge x⟹x\le x$
• Transitive: $\forall x,y,z\in X:(x\le y)\wedge (y\le z)⟹x=x\wedge y=x\wedge (y\wedge z)=(x\wedge y)\wedge z=x\wedge z⟹x\le z$
• Antisymmetric: $\forall x,y\in X:(x\le y)\wedge (y\le x)⟹x=x\wedge y=y\wedge x=y$

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 02