Semilattice

Remarks

The two forms are equivalent. A join-semilattice gives the operation $x \cdot y = x \vee y$. Conversely, a semilattice $(S, \cdot)$ becomes a join-semilattice for the order $x \le y \iff x \cdot y = y$, and a meet-semilattice for the order $x \le y \iff x \cdot y = x$. So a semilattice is a commutative semigroup in which every element is idempotent. Example: the finite subsets of $\N$, ordered by $\subseteq$, form a join-semilattice, since the union of two finite sets is finite and is their join. Example: $\set{a, b, 1}$ with $a < 1$ and $b < 1$ is a join-semilattice with $a \vee b = 1$. It is not a meet-semilattice, since $a$ and $b$ have no common lower bound. Non-example: a two-element antichain $\set{a, b}$ is not a join-semilattice, since $a$ and $b$ have no upper bound.