Editing Semilattice (section)

==Free semilattices==

This section presupposes some knowledge of [[category theory]]. In various situations, [[free object|free]] semilattices exist. For example, the [[forgetful functor]] from the category of join-semilattices (and their homomorphisms) to the [[category theory|category]] of sets (and functions) admits a [[adjoint functors|left adjoint]]. Therefore, the free join-semilattice {{math|1='''F'''(''S'')}} over a set {{math|1=''S''}} is constructed by taking the collection of all non-empty ''finite'' [[subset]]s of {{math|1=''S'',}} ordered by subset inclusion. Clearly, {{math|1=''S''}} can be embedded into {{math|1='''F'''(''S'')}} by a mapping {{math|1=''e''}} that takes any element {{math|1=''s''}} in {{math|1=''S''}} to the singleton set {{math|1={''s''}.}} Then any function {{math|1=''f''}} from a {{math|1=''S''}} to a join-semilattice {{math|1=''T''}} (more formally, to the underlying set of {{math|1=''T''}}) induces a unique homomorphism {{math|1=''f' ''}} between the join-semilattices {{math|1='''F'''(''S'')}} and {{math|1=''T'',}} such that {{math|1=''f'' = ''f' '' ○ ''e''.}} Explicitly, {{math|1=''f' ''}} is given by <math display=inline>f'(A) = \bigvee\{f(s) | s \in A\}.</math> Now the obvious uniqueness of {{math|1=''f' ''}} suffices to obtain the required adjunction—the morphism-part of the functor {{math|1='''F'''}} can be derived from general considerations (see [[adjoint functors]]). The case of free meet-semilattices is dual, using the opposite subset inclusion as an ordering. For join-semilattices with bottom, we just add the empty set to the above collection of subsets.

In addition, semilattices often serve as generators for free objects within other categories. Notably, both the forgetful functors from the category of [[complete Heyting algebra|frames]] and frame-homomorphisms, and from the category of distributive lattices and lattice-homomorphisms, have a left adjoint.