Functorality of Green's relations

Green's relations are part of the relationship between order theory and monoid theory. Green's relations can be expressed in category theory as functors from the categories of monoids to the category of preorders, both of which are full subcategories of the category of categories $Cat$.

Theorem. Green's preorders $\subseteq_L, \subseteq_R, \subseteq_J$ are forgetful functors from the category of monoids to the categories of preorders. \[ \subseteq_L : Mon \to Ord \] \[ \subseteq_R : Mon \to Ord \] \[ \subseteq_J : Mon \to Ord \] Proof. (1) suppose that that $a \subseteq_L b$ then $\exists x : xa = b$ which implies that $f(x)f(a) = f(b)$. This implies that $f(a) \subseteq_L f(b)$ by $f(x)$.

(2) similarly, if $a \subseteq_R b$ then $\exists y: ay = b$ which implies that $f(a)f(y) = f(b)$. This implies that $f(a) \subseteq_R f(b)$ by $f(y)$.

(3) finally, by combining the two we have that $a \subseteq_J b$ then $\exists x,y : xay = b$. This implies that $f(x)f(a)f(y) = f(b)$ which implies that $f(a) \subseteq f(b)$ by $f(x)$ and $f(y)$. $\square$

Green's preorders are functors from the category of monoids to the category of preorders, and Green's relations are as well. The only difference is that Green's relations are always symmetric.

Theorem. Green's relations $L,R,J,D,H$ are functors from the category of monoids to the category of preorders.

Proof. (1) suppose that $a \text{ L } b$ then $a \subseteq_L b$ and $b \subseteq_L a$ so by functoriality $f(a) \subseteq_L f(b)$ and $f(b) \subseteq_L f(a)$ which implies that $f(a) \text{ L } f(b)$. The same applies for $R$ and $J$.

(2) suppose that $a \text { H } b$ then $a \text{ L } b$ and $a \text{ R } b$. By part (1) we have that this implies $f(a) \text{ L } f(b)$ and $f(a) \text{ R } f(b)$. By combination this implies $f(a) \text{ H } f(b)$.

(3) finalyl suppose that $a \text{ D } b$ then because $D$ is defined by transitive closure this implies that there is a chain $a \text{ L } x_1 \text{ R } ... \text{ L } x_n \text{ R } b$. Then we can apply $f$ to this chain of relations to get $f(a) \text{ L } f(x_1) \text{ R} ... \text{ L } f(x_n) \text{ R} f(b)$. This implies that $f(a) \text{ D } f(b)$. $\square$

Green's preorders can be defined as the action preorders of monoid actions, but this is not functorial because each monoid has a different topos of monoid actions, so there is no single output category to define a functor for. So we are going to have make do with the functorality of Green's relations for now.

These theorems can be used as a foundation of a number of more advanced constructions in semigroup theory. For example, we can use this to show that monotone maps reflect ideals from which it follows that semigroup morphism reflect ideals as well. That ring maps reflect ideals immediately follows.