Editing Lattice (order) (section)

== Bounded lattice ==
A '''bounded lattice''' is a lattice that additionally has a {{dfnil|greatest element}} (also called {{dfni|maximum}}, or {{dfni|top}} element, and denoted by <math>1,</math> or {{nowrap|by <math>\top</math>)}} and a {{dfnil|least element}} (also called {{dfni|minimum}}, or {{dfni|bottom}}, denoted by <math>0</math> or by {{nowrap|<math>\bot</math>),}} which satisfy
<math display=block>0 \leq x \leq 1 \;\text{ for every } x \in L.</math>

A bounded lattice may also be defined as an algebraic structure of the form <math>(L, \vee, \wedge, 0, 1)</math> such that <math>(L, \vee, \wedge)</math> is a lattice, <math>0</math> (the lattice's bottom) is the [[identity element]] for the join operation <math>\vee,</math> and <math>1</math> (the lattice's top) is the identity element for the meet operation <math>\wedge.</math><math display=block>a \vee 0 = a</math><math display=block>a \wedge 1 = a</math>

It can be shown that a partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet.

Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by <math display="inline">1 = \bigvee L = a_1 \lor \cdots \lor a_n</math> (respectively <math display="inline">0 = \bigwedge L = a_1 \land \cdots \land a_n</math>) where <math>L = \left\{a_1, \ldots, a_n\right\}</math> is the set of all elements.