# Examples of cancellative normal semigroups

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?

• AFAIK, positive braids have this property. – Alex Degtyarev Jan 17 '14 at 9:21
• Is there any reason to worry about "normal" cancellative semigroups? Seems this definition is abnormal – Victor Jan 19 '14 at 4:57
• @Victor They are inner automorphism invariant in their groups of fractions. – Michał Masny Jan 19 '14 at 10:06

$G\times\{1,2,3,\dots\}$, where $G$ is a nonabelian group.