semilattice
Sign in to saveAlso known as Join-semilattice, Meet-semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
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Encyclopedic overview
13 sectionsContents
- Order-theoretic definition
- Algebraic definition
- Connection between the two definitions
- Examples
- Semilattice morphisms
- Equivalence with algebraic lattices
- Distributive semilattices
- Complete semilattices
- Free semilattices
- See also
- Notes
- References
- External links
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order.
Excerpted from Wikipedia’s “semilattice” article, available under the CC BY-SA 4.0 licence.