Examples
Semilattices are employed to construct other order structures, or in conjunction with other completeness properties.
- A 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 elements of an algebraic lattice, under the induced partial ordering, form a bounded join-semilattice.
- Any tree structure (with the root as the least element) is a meet-semilattice. Consider for example the set of finite words over some alphabet, ordered by the prefix ordering. It has a least but no greatest element: the root is the meet of all other elements.
- A Scott domain is a meet-semilattice.
- Membership in any set L can be taken as a model of a semilattice with base set L, because a semilattice captures the essence of set extensionality. Let a∧b denote a∈L & 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 ∧ assure (1), idempotence, (2). This semilattice is the free semilattice over L. It is not bounded by 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.
Read more about this topic: Semilattice
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