Examples of cancellative normal semigroups

Question

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?

Answer 1

I've never seen the term 'normal' being used in the way of the OP. A semigroup $S$ (written multiplicatively) with the property that $xS = Sx$ for all $x \in S$ is commonly called either duo or normalizing (see also this answer). Cancellative, non-commutative examples abound 'in nature'. Here is a short list:

1. Any group (either commutative or not) is a cancellative, duo semigroup.

2. If $(G, \preceq)$ is a totally ordered group (either commutative or not) with identity element $1_G$, then $G^+ := \{x \in G \colon 1_G \prec x\}$ is a cancellative, duo subsemigroup of $G$. For the duoness part, it suffices to note that, for all $a, b \in G^+$, we have $1_G = aa^{-1} \prec aba^{-1} \in G^+$ and, in a similar way, $a^{-1}ba \in G^+$. (This example is already mentioned by user46855 in their answer.)

3. The multiplicative monoid $R^\bullet$ of the non-zero elements of a left (or right) discrete valutation domain $R$ is a strongly Archimedean, cancellative, duo semigroup ('strongly Archimedean' means that, for every $a \in R^\bullet$, there exists an integer $n \ge 1$ such that any product of any $n$ non-zero non-units of $R$ is divisible by $a$ in $R^\bullet$). One special case is given by the ring of formal power series in one variable $x$ over a skew field $D$, with multiplication twisted by a ring automorphism $\sigma$ of $D$ in such a way that $ax = x\sigma(a)$ for every $a \in D$; see Exercise 19.7 in the 2003 edition of Lam's Exercises in Classical Ring Theory. (This example is also already mentioned by user46855 in their answer.)

4. Any direct product of cancellative, duo semigroups is itself cancellative and duo, and it is commutative if and only if each factor in the direct product is (cf. Anton Klyachko's answer).

Answer 2

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

Answer 3

positive cone of (nonabelian) totally ordered groups (like the multiplicative group in Hilbert's ordered skew field to show that Pappus theorem is not provable in ordered affine geometry). More examples are given by skew polinomial (and formal power series) rings with respect to a (nonidentity) automorphism of the base (skew) field of coefficients. See also Cohn, free rings and their relations, and the more modern version, Free Ideal Rings and Localization in General Rings (especially section 0.7 and final notes to chapter 0).