Editing Semilattice (section)

==Semilattice morphisms==

The above algebraic definition of a semilattice suggests a notion of [[morphism]] between two semilattices. Given two join-semilattices {{math|1=(''S'', ∨)}} and {{math|1=(''T'', ∨)}}, a [[homomorphism]] of (join-) semilattices is a function {{math|1=''f'': ''S'' → ''T''}} such that

:{{math|1=''f''(''x'' ∨ ''y'') = ''f''(''x'') ∨ ''f''(''y'').}}

Hence {{math|1=''f''}} is just a homomorphism of the two [[semigroups]] associated with each semilattice. If {{math|1=''S''}} and {{math|1=''T''}} both include a least element 0, then {{math|1=''f''}} should also be a [[monoid]] homomorphism, i.e. we additionally require that

: {{math|1=''f''(0) = 0.}}

In the order-theoretic formulation, these conditions just state that a homomorphism of join-semilattices is a function that preserves binary joins and least elements, if such there be. The obvious dual—replacing {{math|1=∧}} with {{math|1=∨}} and 0 with 1—transforms this definition of a join-semilattice homomorphism into its meet-semilattice equivalent.

Note that any semilattice homomorphism is necessarily [[monotone function|monotone]] with respect to the associated ordering relation. For an explanation see the entry preservation of limits.