# a semilattice is a commutative band

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 $x\wedge y=xy$.

Title a semilattice is a commutative band ASemilatticeIsACommutativeBand 2013-03-22 12:57:28 2013-03-22 12:57:28 mclase (549) mclase (549) 6 mclase (549) Proof msc 20M99 msc 06A12 Lattice