## Lattice Isomorphism Theorem for semigroups

Let $S$ be a semigroup and let $\sigma$ be a congruence on $S$. Let $\mathsf{Q}_\sigma(S)$ denote the set of all congruences on $S$ which contain $\sigma$, and let $\mathsf{Q}(S/\sigma)$ denote the set of all congruences on $S/\sigma$. Note that both $\mathsf{Q}_\sigma(S)$ and $\mathsf{Q}(S/\sigma)$ are partially ordered by $\subseteq$. Show that in fact they are isomorphic as partially ordered sets.

Given $\rho \in \mathsf{Q}_\sigma(S)$, define a relation $\overline{\rho}$ on $S/\sigma$ by $[x]_\sigma \overline{\rho} [y]_\sigma$ if and only if there exist $x^\prime \sigma x$ and $y^\prime \sigma y$ such that $x^\prime \rho y^\prime$. Since $\sigma \subseteq \rho$, note that in fact $[x]_\sigma \overline{\rho} [y]_\sigma$ if and only if $x \rho y$.

We claim that $\overline{\rho}$ is a congruence on $S/\sigma$. Since $\rho$ is reflextive, we have $x \rho x$ for all $x \in S$, and thus $[x]_\sigma \overline{\rho} [x]_\sigma$ for all $x$. If $[x]_\sigma \overline{\rho} [y]_\sigma$, then $x \rho y$, so that $y \rho x$, and thus $[y]_\sigma \overline{\rho} [y]_\sigma$. If $[x]_\sigma \overline{\rho} [y]_\sigma$ and $[y]_\sigma \overline{\rho} [z]_\sigma$, then $x \rho y \rho z$, so that $x \rho z$ and hence $[x]_\sigma \overline{\rho} [z]_\sigma$. Finally, if $[x]_\sigma \overline{\rho} [y]_\sigma$ and $[z]_\sigma \overline{\rho} [w]_\sigma$, then $x \rho y$ and $z \rho w$, so that $xz \rho yz$, and hence $[x]_\sigma[z]_\sigma \overline{\rho} [y]_\sigma[w]_\sigma$. So $\overline{\rho}$ is a congruence on $S/\sigma$.

Now define $\Psi : \mathsf{Q}_\sigma(S) \rightarrow \mathsf{Q}(S/\sigma)$ by $\Psi(\rho) = \overline{\rho}$. We claim that $\Psi$ is a poset isomorphism.

Suppose $\Psi(\rho) = \Psi(\tau)$. Now for all $x,y \in S$, we have $x \rho y$ if and only if $[x]_\sigma \overline{\rho} [y]_\sigma$ if and only if $[x]_\sigma \overline{\tau} [y]_\sigma$ if and only if $x \tau y$. So $\rho = \tau$. Thus $\Psi$ is injective.

Now suppose $\tau \in \mathsf{Q}(S/\sigma)$. Define a relation $\rho$ on $S$ by $x \rho y$ if and only if $[x]_\sigma \tau [y]_\sigma$. We claim that $\rho$ is a congruence. Indeed, for all $x$, $[x]_\sigma \tau [x]_\sigma$, so $x \rho x$; if $x \rho y$ then $[x]_\sigma \tau [y]_\sigma$, so that $[y]_\sigma \tau [x]_\sigma$, hence $y \rho x$; if $x \rho y$ and $y \rho z$, then $[x]_\sigma \tau [y]_\sigma \tau [z]_\sigma$, so that $[x]_\sigma \tau [z]_\sigma$, and thus $x \rho z$; and if $x \rho y$ and $z \rho w$, then $[x]_\sigma \tau [y]_\sigma$ and $[z]_\sigma \tau [w]_\sigma$, so $[xz]_\sigma \tau [yw]_\sigma$, and thus $xz \rho yw$. We claim also that $\sigma \subseteq \rho$. Indeed, if $x \sigma y$, then $[x]_\sigma = [y]_\sigma$, so that $[x]_\sigma \tau [y]_\sigma$ and we have $x \rho y$. So in fact $\rho \in \mathsf{Q}_\sigma(S)$. Finally, we claim that $\Psi(\rho) = \tau$; in fact, $[x]_\sigma \Psi(\rho) [y]_\sigma$ if and only if $x \rho y$ if and only if $[x]_\sigma \tau [y]_\sigma$ as desired. So $\Psi$ is surjective.

So $\Psi$ is a bijection. Finally, we claim that $\Psi$ preserves set containment. Indeed, if $\rho \subseteq \tau$, then $[x]_\sigma \Psi(\rho) [y]_\sigma$ implies $x \rho y$, implies $x \tau y$, implies $[x]_\sigma \Psi(\tau) [y]_\sigma$. So $\Psi(\rho) \subseteq \Psi(\tau)$.

Hence $\mathsf{Q}_\sigma(S)$ and $\mathsf{Q}(S/\sigma)$ are isomorphic as partially ordered sets.

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