Editing Semilattice (section)

== Connection between the two definitions ==
An order theoretic meet-semilattice {{math|1=&lang;''S'', ≤&rang;}} gives rise to a [[binary operation]] {{math|1=∧}} such that {{math|1=&lang;''S'', ∧&rang;}} is an algebraic meet-semilattice. Conversely, the meet-semilattice {{math|1=&lang;''S'', ∧&rang;}} gives rise to a [[binary relation]] {{math|1=≤}} that partially orders {{math|1=''S''}} in the following way: for all elements {{math|1=''x''}} and {{math|1=''y''}} in {{math|1=''S'', ''x'' ≤ ''y''}} if and only if {{math|1=''x'' = ''x'' ∧ ''y''.}}

The relation {{math|1=≤}} introduced in this way defines a partial ordering from which the binary operation {{math|1=∧}} may be recovered. Conversely, the order induced by the algebraically defined semilattice {{math|1=&lang;''S'', ∧&rang;}} coincides with that induced by {{math|1=≤.}}

Hence the two definitions may be used interchangeably, depending on which one is more convenient for a particular purpose. A similar conclusion holds for join-semilattices and the dual ordering ≥.