Editing Semilattice (section)

==Order-theoretic definition==

A [[set (mathematics)|set]] {{math|1=''S''}} [[partially ordered set|partially ordered]] by the [[binary relation]] {{math|1=≤}} is a ''meet-semilattice'' if

: For all elements {{math|1=''x''}} and {{math|1=''y''}} of {{math|1=''S''}}, the [[infimum|greatest lower bound]] of the set {{math|1={''x'', ''y''} }} exists.

The greatest lower bound of the set {{math|1={''x'', ''y''} }} is called the [[meet (mathematics)|meet]] of {{math|1=''x''}} and {{math|1=''y'',}} denoted {{math|1=''x'' ∧ ''y''.}}

Replacing "greatest lower bound" with "[[supremum|least upper bound]]" results in the dual concept of a ''join-semilattice''. The least upper bound of {{math|1={''x'', ''y''} }} is called the [[Join (mathematics)|join]] of {{math|1=''x''}} and {{math|1=''y''}}, denoted {{math|1=''x'' ∨ ''y''}}. Meet and join are [[binary operation]]s on {{math|1=''S''.}} A simple [[mathematical induction|induction]] argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non-empty finite suprema (infima).

A join-semilattice is '''bounded''' if it has a [[least element]], the join of the empty set. [[Duality (order theory)|Dually]], a meet-semilattice is '''bounded''' if it has a [[greatest element]], the meet of the empty set.

Other properties may be assumed; see the article on [[completeness (order theory)|completeness in order theory]] for more discussion on this subject. That article also discusses how we may rephrase the above definition in terms of the existence of suitable [[Galois connection]]s between related posets — an approach of special interest for [[category theory|category theoretic]] investigations of the concept.