Semilattice in the context of Distributivity (order theory)

⭐ Core Definition: 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.

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.

↓ Menu

HINT:

👉 Semilattice in the context of Distributivity (order theory)

In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.

↓ Explore More Topics

In this Dossier

⭐ Core Definition: Semilattice
👉 Semilattice in the context of Distributivity (order theory)
Semilattice in the context of Lattice (order)

Semilattice in the context of Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions.

View the full Wikipedia page for Lattice (order)

↑ Return to Menu

Semilattice in the context of Distributivity (order theory)

Semilattice Study page number 1 of 1

Play TriviaQuestions Online!

Skip to study material about Semilattice in the context of "Distributivity (order theory)"

⭐ Core Definition: Semilattice

👉 Semilattice in the context of Distributivity (order theory)

Semilattice in the context of Lattice (order)