This note explains how a semilattice is the same as a commutative band.

Let $S$ be a semilattice, with partial order $<$ and each pair of elements $x$ and $y$ having a greatest lower bound $x \wedge y$ Then it is easy to see that the operation $\wedge$ defines a binary operation on $S$ which makes it a commutative semigroup, and that every element is idempotent since $x \wedge x = x$

Conversely, if $S$ is such a semigroup, define $x \leq y$ iff $x = xy$ Again, it is easy to see that this defines a partial order on $S$ and that greatest lower bounds exist with respect to this partial order, and that in fact $x \wedge y = xy$