3 Basic Properties of Green’s Relations
Let \(e\) be an idempotent element in a semigroup \(S\). An element \(x \in S\) is \(\mathcal{R}\)-below \(e\) if and only if \(x = ex\). Similarly, \(x\) is \(\mathcal{L}\)-below \(e\) if and only if \(x = xe\). It follows that \(x\) is \(\mathcal{H}\)-below \(e\) if and only if both conditions hold, i.e., \(x = ex\) and \(x = xe\).
For the \(\mathcal{R}\)-preorder, if \(x \leq _{\mathcal{R}} e\), then \(x = ez\) for some \(z \in S^1\). Since \(e\) is idempotent, \(e=e^2\), so \(ex = e(ez) = e^2z = ez = x\). Conversely, if \(x=ex\), then \(x \leq _{\mathcal{R}} e\) by definition. The argument for the \(\mathcal{L}\)-preorder is analogous. The statement for \(\mathcal{H}\) follows directly from the definitions.
Green’s relations are preserved under semigroup morphisms. Let \(f: S \to T\) be a semigroup morphism. If two elements \(x, y \in S\) are related by any of Green’s preorders or equivalence relations, then their images \(f(x), f(y)\) are related by the same relation in \(T\).
If \(x \leq _{\mathcal{R}} y\), then \(x = yz\) for some \(z \in S^1\). Applying the morphism \(f\) gives \(f(x) = f(yz) = f(y)f(z)\), so \(f(x) \leq _{\mathcal{R}} f(y)\). This extends to the equivalence relation: if \(x \mathcal{R} y\), then \(x \leq _{\mathcal{R}} y\) and \(y \leq _{\mathcal{R}} x\), which implies \(f(x) \leq _{\mathcal{R}} f(y)\) and \(f(y) \leq _{\mathcal{R}} f(x)\), so \(f(x) \mathcal{R} f(y)\).
A similar argument holds for \(\mathcal{L}\). The preservation of \(\mathcal{J}\) and \(\mathcal{H}\) follows from their definitions. For \(\mathcal{D}\), if \(x \mathcal{D} y\), there exists \(z\) such that \(x \mathcal{R} z\) and \(z \mathcal{L} y\). Then \(f(x) \mathcal{R} f(z)\) and \(f(z) \mathcal{L} f(y)\), which implies \(f(x) \mathcal{D} f(y)\).