a semilattice is a commutative band
This note explains how a semilattice is the same as a commutative band.
Let be a semilattice, with partial order and each pair of elements and having a greatest lower bound
.Then it is easy to see that the operation
defines a binary operation
on which makes it a commutative semigroup, and that every element is idempotent
since .
Conversely, if is such a semigroup, define iff . Again, it is easy to see that this defines a partial order on , and that greatest lower bounds exist with respect to this partial order, and that in fact .