Equivalence of Definitions of Lattice Isomorphism

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Theorem

Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.


The following definitions of the concept of Lattice Isomorphism are equivalent:

Definition 1

Let $\phi: L_1 \to L_2$ be a (lattice) homomorphism.


We say $\phi: L_1 \to L_2$ is a lattice isomorphism if and only if $\phi : A_1 \to A_2$ is a bijection.


Definition 2

Let $\phi: A_1 \to A_2$ be a mapping.


We say $\phi : L_1 \to L_2$ is a lattice isomorphism if and only if $\phi : L_1 \to L_2$ is an order isomorphism.


Proof

Definition 1 implies Definition 2

Let $\phi : L_1 \to L_2$ be a bijective lattice homomorphism.


From Inverse of Bijective Lattice Homomorphism is Bijective Lattice Homomorphism:

$\phi^{-1}: L_2 \to L_1$ is a bijective lattice homomorphism.


From Lattice Homomorphism is Order-Preserving:

$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ and $\phi^{-1} : \struct{A_2, \preceq_2} \to \struct{A_1, \preceq_1}$ are order-preserving


Hence $\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ is an order isomorphism by definition.

$\Box$

Definition 2 implies Definition 1

Let $\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ be an order isomorphism.


By definition of order isomorphism:

$\phi : \struct{A_1, \preceq_1} \to \struct{A_2, \preceq_2}$ is an order-preserving bijection


Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.

From Inverse of Order Isomorphism is Order Isomorphism:

$\phi^{-1} : \struct{A_2, \preceq_2} \to \struct{A_1, \preceq_1}$ is an order isomorphism


$\phi$ Satisfies Join Morphism Property

Let:

$x, y \in A_1$


We have:

\(\ds x, y\) \(\le\) \(\ds x \vee_1 y\) Definition of Supremum
\(\ds \leadsto \ \ \) \(\ds \map \phi x, \map \phi y\) \(\le\) \(\ds \map \phi {x \vee_1 y}\) Definition of Order-Preserving Mapping
\(\ds \leadsto \ \ \) \(\ds \map \phi x \vee_2 \map \phi y\) \(\le\) \(\ds \map \phi {x \vee_1 y}\) Definition of Supremum
\(\ds \leadsto \ \ \) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) \(\le\) \(\ds \map {\phi^{-1} } {\map \phi {x \vee_1 y} }\) Definition of Order-Preserving Mapping
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) \(\le\) \(\ds x \vee_1 y\) Definition of Inverse Mapping


Also we have:

\(\ds \map \phi x, \map \phi y\) \(\le\) \(\ds \map \phi x \vee_2 \map \phi y\) Definition of Supremum
\(\ds \leadsto \ \ \) \(\ds \map {\phi^{-1} } {\map \phi x}, \map {\phi^{-1} } {\map \phi y}\) \(\le\) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) Definition of Order-Preserving Mapping
\(\ds \leadsto \ \ \) \(\ds x, y\) \(\le\) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) Definition of Inverse Mapping
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds x \vee_1 y\) \(\le\) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) Definition of Supremum


Hence:

\(\ds x \vee_1 y\) \(=\) \(\ds \map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y}\) Ordering Axiom $(3)$: Antisymmetry applied to $(1)$ and $(2)$
\(\ds \leadsto \ \ \) \(\ds \map \phi {x \vee_1 y}\) \(=\) \(\ds \map \phi {\map {\phi^{-1} } {\map \phi x \vee_2 \map \phi y} }\)
\(\ds \) \(=\) \(\ds \map \phi x \vee_2 \map \phi y\) Definition of Inverse Mapping


It follows:

$(3):\quad \forall x, y \in A_1 : \map \phi {x \vee_1 y} = \map \phi x \vee_2 \map \phi y$

That is, $\phi$ satisfies the join morphism property.

$\Box$


$\phi$ Satisfies Meet Morphism Property

The dual statement of $(3)$ is:

$(4):\quad \forall x, y \in A_1 : \map \phi {x \wedge_1 y} = \map \phi x \wedge_2 \map \phi y$

by Dual Pairs (Order Theory).


We have $(4)$ holds from Duality Principle.


That is, $\phi$ satisfies the meet morphism property.

$\Box$


It follows that $\phi$ is a bijective lattice homomorphism by definition.

$\blacksquare$

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