$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$


$A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$


It is trivial to see the containment $SH \leq HS$. Taking a simple example of rings also proves $HS \neq SH$. Currently I am stuck with proving $SH(K) = HS(K)$ where K is a monounary algebra $(Z,successor).$


This question might sound belonging to stack exchange, but there isn't a large community of universal algebraists over there.



H S class operator and its equality
I'm assuming that $(Z,successor)$ represents the set of integers with the successor function $s\colon Z\to Z$, $s(a)=a+1$. 


I think the following works: we want to show that any homomorphic image of a subalgebra of $(Z,s)$ is a subalgebra of a homomorphic image of $(Z,s)$. A subalgebra of $(Z,s)$ is either the whole of $Z$ (in which case we don't have a problem showing any homomorphic image lies in $SH(Z)$) or else it is of the form $\{k\in Z\mid k\geq n\_0\}$ for some integer $n\_0$. These are all isomorphic to $(\mathbb{N},s)$, so we just need to consider homomorphic images of $(\mathbb{N},s)$. 


Let $\Phi$ be a congruence on $(\mathbb{N},s)$, and let $\Psi = \Phi\cup\{(a,a)\mid a\in\mathbb{Z}, a\lt 0\}$ (my natural numbers include $0$). Then it is easy to verify that $\Psi$ is a congruence on $\mathbb{Z}$, and that if $x\in\mathbb{Z}$ and $a\in\mathbb{Z}$, $a\lt 0$, then $[x]=[a]$ if and only if $x=a$. Thus, the subalgebra of $Z/\Psi$ consisting of the equivalence classes represented by nonnegative integers form a subalgebra that is isomorphic to $\mathbb{N}/\Phi$, showing that every homomorphic image of a subalgebra may be realized as a subalgebra of a homomorphic image in this situation.


