Editing Lattice (order) (section)

== Definition ==
A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.

=== As partially ordered set ===
A [[partially ordered set]] (poset) <math>(L, \leq)</math> is called a '''lattice''' if it is both a join- and a meet-[[semilattice]], i.e. each two-element subset <math>\{ a, b \} \subseteq L</math> has a [[Join (mathematics)|join]] (i.e. least upper bound, denoted by <math>a \vee b</math>) and [[Duality (order theory)|dually]] a [[Meet (mathematics)|meet]] (i.e. greatest lower bound, denoted by <math>a \wedge b</math>). This definition makes <math>\,\wedge\,</math> and <math>\,\vee\,</math> [[binary operation]]s. Both operations are monotone with respect to the given order: <math>a_1 \leq a_2</math> and <math>b_1 \leq b_2</math> implies that <math>a_1 \vee b_1 \leq a_2 \vee b_2</math> and <math>a_1 \wedge b_1 \leq a_2 \wedge b_2.</math>

It follows by an [[Mathematical induction|induction]] argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; see ''[[Completeness (order theory)]]'' for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable [[Galois connection]]s between related partially ordered sets—an approach of special interest for the [[category theoretic]] approach to lattices, and for [[formal concept analysis]].

Given a subset of a lattice, <math>H \subseteq L,</math> meet and join restrict to [[partial function]]s – they are undefined if their value is not in the subset <math>H.</math> The resulting structure on <math>H</math> is called a '''{{visible anchor|partial lattice}}'''. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.{{sfn|Grätzer|2003|p=[https://books.google.com/books?id=SoGLVCPuOz0C&pg=PA52 52]}}

=== As algebraic structure ===
A '''lattice''' is an [[algebraic structure]] <math>(L, \vee, \wedge)</math>, consisting of a set <math>L</math> and two binary, commutative and associative [[Operation (mathematics)|operations]] <math>\vee</math> and <math>\wedge</math> on <math>L</math> satisfying the following axiomatic identities for all elements <math>a, b \in L</math> (sometimes called {{em|absorption laws}}):
<math display=block>a \vee (a \wedge b) = a</math>
<math display=block>a \wedge (a \vee b) = a</math>

The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.<ref>{{harvnb|Birkhoff|1948|p=[https://archive.org/details/in.ernet.dli.2015.166886/page/n35/mode/2up 18]}}. "since <math>a = a \vee (a \wedge (a \vee a)) = a \vee a</math> and dually". Birkhoff attributes this to {{harvnb|Dedekind|1897|p=[https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf#page=10 8]}}</ref> These are called {{em|idempotent laws}}.
<math display=block>a \vee a = a</math>
<math display=block>a \wedge a = a</math>

These axioms assert that both <math>(L, \vee)</math> and <math>(L, \wedge)</math> are [[semilattice]]s. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the [[Duality (order theory)|dual]] of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same [[partial order]].

=== Connection between the two definitions ===
An order-theoretic lattice gives rise to the two binary operations <math>\vee</math> and <math>\wedge.</math> Since the commutative, associative and absorption laws can easily be verified for these operations, they make <math>(L, \vee, \wedge)</math> into a lattice in the algebraic sense.

The converse is also true. Given an algebraically defined lattice <math>(L, \vee, \wedge),</math> one can define a partial order <math>\leq</math> on <math>L</math> by setting
<math display=block>a \leq b \text{ if } a = a \wedge b, \text{ or }</math>
<math display=block>a \leq b \text{ if } b = a \vee b,</math>
for all elements <math>a, b \in L.</math> The laws of absorption ensure that both definitions are equivalent:
<math display=block>a = a \wedge b \text{ implies } b = b \vee (b \wedge a) = (a \wedge b) \vee b = a \vee b</math>
and dually for the other direction.

One can now check that the relation <math>\le</math> introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations <math>\vee</math> and <math>\wedge.</math>

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.