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 $a,b\in L$,

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

• $\varphi (a\vee b)=\varphi (a)\vee \varphi (b)$.

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.

• 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.

• 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.

• One can show that every Boolean algebra $B$ can be embedded into the power set of some set $S$. That is, there is a one-to-one 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	2013-03-22 15:41:31
Last modified on	2013-03-22 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