Editing Semilattice (section)

==Examples==

Semilattices are employed to construct other order structures, or in conjunction with other completeness properties.
* A [[lattice (order)|lattice]] is both a join- and a meet-semilattice. The interaction of these two semilattices via the [[absorption law]] is what truly distinguishes a lattice from a semilattice.
* The [[compact element]]s of an algebraic [[lattice (order)|lattice]], under the induced partial ordering, form a bounded join-semilattice.
* By induction on the number of elements, any non-empty finite meet semilattice has a least element and any non-empty finite join semilattice has a greatest element. (In neither case will the semilattice necessarily be bounded.)
* A [[totally ordered set]] is a [[distributive lattice]], hence in particular a meet-semilattice and join-semilattice: any two distinct elements have a greater and lesser one, which are their meet and join.
** A [[well-ordered set]] is further a ''bounded'' join-semilattice, as the set as a whole has a least element, hence it is bounded.
*** The [[Natural number#Order|natural numbers]] <math>\mathbb{N}</math>, with their usual order {{math|1=≤,}} are a bounded join-semilattice, with least element 0, although they have no greatest element: they are the smallest infinite well-ordered set.
* Any single-rooted [[Tree (set theory)|tree]] (with the single root as the least element) of height <math>\leq \omega</math> is a (generally unbounded) meet-semilattice. Consider for example the set of finite words over some alphabet, ordered by the [[prefix order]]. It has a least element (the empty word), which is an annihilator element of the meet operation, but no greatest (identity) element.
* A [[Scott domain]] is a meet-semilattice.
* Membership in any set {{math|1=''L''}} can  be taken as a [[model theory|model]] of a semilattice with base set {{math|1=''L'',}} because a semilattice captures the essence of set [[extensionality]]. Let {{math|1=''a'' ∧ ''b''}} denote {{math|1=''a'' ∈ ''L''}} & {{math|1=''b'' ∈ ''L''.}} Two sets differing only in one or both of the:
# Order in which their members are listed;
# Multiplicity of one or more members,
:are in fact the same set. Commutativity and associativity of {{math|1=∧}} assure (1), [[idempotence]], (2). This semilattice is the [[free semilattice]] over {{math|1=''L''.}}  It is not bounded by {{math|1=''L'',}} because a set is not a member of itself.
* Classical extensional [[mereology]] defines a join-semilattice, with join read as binary fusion. This semilattice is bounded from above by the world individual.
* Given a set {{math|1=''S'',}} the collection of partitions <math> \xi </math> of {{math|1=''S''}} is a join-semilattice. In fact, the partial order is given by <math> \xi \leq \eta </math> if <math> \forall Q \in \eta, \exists P \in \xi </math> such that <math> Q \subset P </math> and the join of two partitions is given by <math> \xi \vee \eta = \{ P \cap Q \mid P \in \xi \ \& \ Q \in \eta \} </math>.  This semilattice is bounded, with the least element being the singleton partition <math> \{ S \} </math>.