a semilattice is 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 .
|Title||a semilattice is a commutative band|
|Date of creation||2013-03-22 12:57:28|
|Last modified on||2013-03-22 12:57:28|
|Last modified by||mclase (549)|