Join-semilattice in the context of Nonempty set

In mathematics, specifically order theory, the join of a subset ${\displaystyle S}$ of a partially ordered set ${\displaystyle P}$ is the supremum (least upper bound) of ${\displaystyle S,}$ denoted ${\textstyle \bigvee S,}$ and similarly, the meet of ${\displaystyle S}$ is the infimum (greatest lower bound), denoted ${\textstyle \bigwedge S.}$ In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion.

A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.