Editing Lattice (order) (section)

=== 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]}}