Editing Lattice (order) (section)

== Morphisms of lattices ==
[[File:Monotonic but nonhomomorphic map between lattices.gif|thumb|'''Pic.&nbsp;9:''' Monotonic map <math>f</math> between lattices that preserves neither joins nor meets, since <math>f(u) \vee f(v) = u^{\prime} \vee u^{\prime}= u^{\prime}</math> <math>\neq</math> <math>1^{\prime} = f(1) = f(u \vee v)</math> and <math>f(u) \wedge f(v) = u^{\prime} \wedge u^{\prime} = u^{\prime}</math> <math>\neq</math> <math>0^{\prime} = f(0) = f(u \wedge v).</math>]]
The appropriate notion of a [[morphism]] between two lattices flows easily from the [[#Lattices as algebraic structures|above]] algebraic definition. Given two lattices <math>\left(L, \vee_L, \wedge_L\right)</math> and <math>\left(M, \vee_M, \wedge_M\right),</math> a '''lattice homomorphism''' from ''L'' to ''M'' is a function <math>f : L \to M</math> such that for all <math>a, b \in L:</math>
<math display=block>f\left(a \vee_L b\right) = f(a) \vee_M f(b), \text{ and }</math>
<math display=block>f\left(a \wedge_L b\right) = f(a) \wedge_M f(b).</math>

Thus <math>f</math> is a [[homomorphism]] of the two underlying [[semilattice]]s. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a '''bounded-lattice homomorphism''' (usually called just "lattice homomorphism") <math>f</math> between two bounded lattices <math>L</math> and <math>M</math> should also have the following property:
<math display=block>f\left(0_L\right) = 0_M, \text{ and }</math>
<math display=block>f\left(1_L\right) = 1_M.</math>

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily [[Monotone function|monotone]] with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic.&nbsp;9), although an [[Monotonic function|order-preserving]] [[bijection]] is a homomorphism if its [[Inverse function|inverse]] is also order-preserving.

Given the standard definition of [[isomorphism]]s as invertible morphisms, a {{dfni|lattice isomorphism}} is just a [[bijective]] lattice homomorphism. Similarly, a {{dfni|lattice endomorphism}} is a lattice homomorphism from a lattice to itself, and a {{dfni|lattice automorphism}} is a bijective lattice endomorphism.  Lattices and their homomorphisms form a [[Category theory|category]].

<!-- not sure if this is the best place for this -->Let <math>\mathbb{L}</math> and <math>\mathbb{L}'</math> be two lattices with '''0''' and '''1'''. A homomorphism from <math>\mathbb{L}</math> to <math>\mathbb{L}'</math> is called '''0''','''1'''-''separating'' [[if and only if]] <math>f^{-1}\{f(0)\} = \{0\}</math> (<math>f</math> separates '''0''') and <math>f^{-1}\{f(1)\}=\{1\}</math> (<math>f</math> separates 1).