TITLE: Can a Shelah semigroup be commutative?
QUESTION [7 upvotes]: A semigroup $S$ is called 
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$.

Question. Can an infinite Shelah semigroup be commutative?

This problem was motivated by the following results:
Theorem (Shelah, 1980). For any infinite cardinal $\lambda$ with $\lambda^+=2^\lambda$ there exists a group $G$ of cardinality $|G|=\lambda^+$, which is a $6640$-Shelah semigroup. 
Corollary. Under CH there exists a Shelah semigroup of cardinality $\aleph_1$.
Theorem (Protasov, 2010). Each countable Shelah semigroup is finite.
Theorem (folklore?). A commutative group is finite iff it is a Shelah semigroup.
Proposition (@YCor, 2018). A group is finite iff it is a 3-Shelah semigroup.
Theorem (Todorcevic, 1987). There is a commutative binary operation $\cdot:X\times X\to X$ of a set $X$ of cardinality $|X|=\aleph_1$ such that $X=A^2:=\{ab:a,b\in A\}$ for any uncountable subset $A\subset X$.

Added in Edit, after reading the answer of Keith Kearnes who referred to the paper of Ralph McKenzie who studied Jonsson semigroups.
Let us recall that a semigroup $S$ is Jonsson if $S=\bigcup_{n\in\mathbb N}A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$. It is clear that each Shelah semigroup is Jonsson.

Theorem (McKenzie, 1971). A Jonsson semigroup $S$ of infinite cardinality $\kappa$ is a non-commutative group if $\mathrm{cf}(\kappa)>\omega$ or $2^{<\kappa}\le\kappa$. 

McKenzie asked in his paper if this theorem remains true without set-theoretic assumptions.

Problem (McKenzie, 1971). Is each infinite Jonsson semigroup a group?

Is this problem of McKenzie still open?

REPLY [12 votes]: An infinite Shelah semigroup must be a Jonsson semigroup (meaning that it is an infinite semigroup whose proper subsemigroups have lesser power). Therefore the following paper answers the question asked on this page:
McKenzie, Ralph 
On semigroups whose proper subsemigroups have lesser power. 
Algebra Universalis 1 (1971), no. 1, 21-25.
McKenzie proves that 
(1) the only commutative Jonsson semigroups are the generalized cyclic groups, and
(2) under GCH every Jonsson semigroup is the underlying semigroup of a group.
To reiterate and overexplain: infinite + Shelah + commutative implies infinite + Jonsson + commutative implies generalized cyclic. But the generalized cyclic groups are not Shelah. Thus infinite + Shelah + commutative semigroups do not exist.