Semilattice - Order-theoretic Definition

Order-theoretic Definition

A set S partially ordered by the binary relation ≤ is a meet-semilattice if

For all elements x and y of S, the greatest lower bound of the set {x, y} exists.

The greatest lower bound of the set {x, y} is called the meet of x and y, denoted x ∧ y.

Replacing "greatest lower bound" with "least upper bound" results in the dual concept of a join-semilattice. The least upper bound of {x, y} is called the join of x and y, denoted x ∨ y. Meet and join are binary operations on S. A simple induction argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non-empty finite suprema (infima).

A join-semilattice is bounded if it has a least element, the join of the empty set. Dually, a meet-semilattice is bounded if it has a greatest element, the meet of the empty set.

Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject. That article also discusses how we may rephrase the above definition in terms of the existence of suitable Galois connections between related posets — an approach of special interest for category theoretic investigations of the concept.