Example of an infinite semigroup where $\mathcal{D} \neq \mathcal{J}$

Question

Let $S$ be a semigroup and take elements $a, b$ in $S$. Consider the following two Green's relations

$a\mathcal{J}b \iff SaS = SbS$

$a\mathcal{D}b \iff \exists c \text{ such that } aS = cS \text{ and } Sb = Sc$

In any finite semigroup we have $\mathcal{D} = \mathcal{J}$, so what are some examples of infinite semigroups where they aren't equal?

Answer 1

In the semigroup of matrices of the form $\pmatrix{a & 0\\ b & 1}$, where $a$ and $b$ are positive rational numbers, the relation $\cal D$ is the equality, but the relation $\cal J$ is the universal relation.