If a semigroup satisfies these identities, is it necessarily commutative?

Question

Suppose a semigroup $X$ satisfies the following identities.

$$xya\equiv yxa,\quad axy \equiv ayx$$

Without assuming anything further, can we deduce that $X$ satisfies $xy \equiv yx$? In particular, we're not allowed to assume the existence of a neutral element.

Motivation. Let $X$ denote magma. Then for each $x \in X$, we get a left action $\lambda_x : X \rightarrow X$ and a right action $\rho_x : X \rightarrow X$ defined as follows.

$$\lambda_x(a) = xa,\quad \rho_x(a)=ax$$

The statement that $X$ is a semigroup can be expressed: for all $x,y \in X$ we have that $\lambda_x$ commutes with $\rho_y$. The statement that $X$ satisfies $xya\equiv yxa$ can be expressed: for all $x,y \in X$, we have that $\lambda_x$ commutes with $\lambda_y$. Similarly for the dual statement.

Thus, the semigroups satisfying the aforementioned identities are precisely the magmas in which all left and right actions commute. I am curious as to whether this implies that the law of composition is itself commutative.

Answer 1

Minimal counterexample: the semigroup $S$ with zero and two generators $a$ and $b$ presented by the relations $aa = bb = aba = bab = 0$. Then $S = \{a, b, ab, ba, 0\}$ and $ab \not= ba$. It satisfies the identity $xyz = 0$ and consequently the identities $xyz = yxz$ and $zxy = zyx$.

Answer 2

The answer is no. Consider the semigroup $S$ of all nonempty words formed from the letters $a$ and $b$ together with the concatenation operation.

Declare two words $w$ and $w'$ to be equivalent, $w \sim w'$, if either $w = w'$ or $w$ and $w'$ each have length at least 3 and they have the same number of $a$'s and $b$'s. Then it is easily verified that the concatenation operation is compatible with $\sim$, so $T = S/\sim$ is itself a semigroup.

Now $T$ satisfies the two relations mentioned in the problem, since they can only possibly concern words of length at least 3. But the words $ab$ and $ba$ are not equal in $T$, so $T$ is not commutative.