lattice homomorphism
Let $L$ and $M$ be lattices. A map $\varphi $ from $L$ to $M$ is called a lattice homomorphism^{} if $\varphi $ respects meet and join. That is, for $a,b\in L$,

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$\varphi (a\wedge b)=\varphi (a)\wedge \varphi (b)$, and

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$\varphi (a\vee b)=\varphi (a)\vee \varphi (b)$.
From this definition, one also defines lattice isomorphism, lattice endomorphism, lattice automorphism respectively, as a bijective^{} lattice homomorphism, a lattice homomorphism into itself, and a lattice isomorphism onto itself.
If in addition $L$ is a bounded lattice^{} with top $1$ and bottom $0$, with $\varphi $ and $M$ defined as above, then $\varphi (a)=\varphi (1\wedge a)=\varphi (1)\wedge \varphi (a)$, and $\varphi (a)=\varphi (0\vee a)=\varphi (0)\vee \varphi (a)$ for all $a\in L$. Thus $L$ is mapped onto a bounded^{} sublattice $\varphi (L)$ of $M$, with top $\varphi (1)$ and bottom $\varphi (0)$.
If both $L$ and $M$ are bounded with lattice homomorphism $\varphi :L\to M$, then $\varphi $ is said to be a $\{0,1\}$lattice homomorphism if $\varphi (1)$ and $\varphi (0)$ are top and bottom of $M$. In other words,
$$\varphi ({1}_{L})={1}_{M}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}\varphi ({0}_{L})={0}_{M},$$ 
where ${1}_{L},{1}_{M},{0}_{L},{0}_{M}$ are top and bottom elements of $L$ and $M$ respectively.
Remarks.

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The idea behind these definitions comes from the idea of a homomorphism^{} between two algebraic systems of the same type. We require the the homomorphism to preserve all finitary operations, including the nullary ones. This means that if the algebraic system contains constants, they need to be preserved under the homomorphism. Thus, if $L$ and $M$ are both bounded lattices, a homomorphism between $L$ and $M$ must preserve $0$ and $1$. Similarly, if $L$ only has $0$ and $M$ is bounded, then a homomorphism between them should preserve $0$ alone.

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In the case of complete^{} lattices, there are operations^{} that are infinitary, so the homomorphism between two complete lattices should preserve the infinitary operations as well. The resulting lattice homomorphism is a complete lattice homomorphism.

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One can show that every Boolean algebra^{} $B$ can be embedded into the power set^{} of some set $S$. That is, there is a onetoone lattice homomorphism $\varphi $ from $B$ into a Boolean subalgebra of ${2}^{S}$ (under the usual set union and set intersection^{} operations) (see link below). If $B$ is in addition a complete lattice^{} and an atomic lattice, then $B$ is lattice isomorphic to ${2}^{S}$ for some set $S$.
Title  lattice homomorphism 
Canonical name  LatticeHomomorphism 
Date of creation  20130322 15:41:31 
Last modified on  20130322 15:41:31 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B05 
Classification  msc 06B99 
Related topic  OrderPreservingMap 
Related topic  RepresentingABooleanLatticeByFieldOfSets 
Defines  lattice isomorphism 
Defines  lattice endomorphism 
Defines  lattice automorphism 
Defines  
Defines  1}latticehomomorphism 