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?

The author constructs some remarkable infinite groups, notably "Jónsson groups'', that is, groups of uncountable cardinality containing no proper subgroups (or better: no proper subsemigroup) of the same cardinality. One construction works for all successor cardinals if we assume the generalized continuum hypothesis; a variant works for $\aleph_1$ with no set-theoretic assumptions.mathscinet.ams.org/mathscinet-getitem?mr=579953 But maybe this refers to being Jonsson, and not the more precise exponent fact? I don't have the paper actually. $\endgroup$ – YCor Oct 30 '18 at 10:374more comments