Cancellation in primely generated refinement monoids

Abstract.

We prove that any primely generated refinement monoid M has separative cancellation, and even strong separative cancellation provided M has no nonzero idempotents. A form of multiplicative cancellation also holds: $na\leq nb$ implies $a\leq b$ for $a,b \in M$ and $n \in \{1,2,3,\ldots\}$. In addition, M is a semilattice in the sense that, given $c_1,c_2 \in M$, there is an element $d \in M$ such that $c_1,c_2 \leq d$ and, for all $a \in M, c_1,c_2 \leq a$ implies $d \leq a$. Finally, we prove that any finitely generated refinement monoid is primely generated; in fact, this holds for any refinement monoid with a set of generators satisfying the descending chain condition.