Editing Semilattice (section)

==Algebraic definition==

A ''meet-semilattice'' is an [[algebraic structure]] <math>\langle S, \land \rangle</math> consisting of a [[set (mathematics)|set]] {{math|1=''S''}} with a [[binary operation]] {{math|1=∧}}, called '''meet''', such that for all members {{math|1=''x'', ''y'',}} and {{math|1=''z''}} of {{math|1=''S'',}} the following [[identity (mathematics)|identities]] hold:

; [[Associativity]]: {{math|1=''x'' ∧ (''y'' ∧ ''z'') = (''x'' ∧ ''y'') ∧ ''z''}}
; [[Commutativity]]: {{math|1=''x'' ∧ ''y'' = ''y'' ∧ ''x''}}
; [[Idempotency]]: {{math|1=''x'' ∧ ''x'' = ''x''}}

A meet-semilattice <math>\langle S, \land \rangle</math> is '''bounded''' if {{math|1=''S''}} includes an [[identity element]] 1 such that {{math|1=''x'' ∧ 1 {{=}} ''x''}} for all {{math|1=''x''}} in {{math|1=''S''.}} 

If the symbol {{math|1=∨}}, called '''join''', replaces {{math|1=∧}} in the definition just given, the structure is called a ''join-semilattice''. One can be ambivalent about the particular choice of symbol for the operation, and speak simply of ''semilattices''.

A semilattice is a [[commutativity|commutative]], [[idempotency|idempotent]] [[semigroup]]; i.e., a commutative [[band (mathematics)|band]]. A bounded semilattice is an idempotent commutative [[monoid]].

A partial order is induced on a meet-semilattice by setting {{math|1=''x'' ≤ ''y''}} whenever {{math|1=''x'' ∧ ''y'' {{=}} ''x''}}. For a join-semilattice, the order is induced by setting {{math|1=''x'' ≤ ''y''}} whenever {{math|1=''x'' ∨ ''y'' {{=}} ''y''}}. In a bounded meet-semilattice, the identity 1 is the greatest element of {{math|1=''S''.}} Similarly, an identity element in a join semilattice is a least element.