I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in S\implies (\exists a,b\in S)\, xa=yb$$ since then for all $x,y$ there exists $a\in S$ such that $xy=ya.$

Could you give me some concrete examples of noncommutative, cancellative on both sides, normal semigroups (non-groups) or some references to examples?