Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.

A function $f:S\to\Bbb{C}$ is **semicharacter** on the semigroup $S$, if it is multiplicative, where $\Bbb{C}$ is complex plane.

Let $S,T$ be two semigroups and $Aut(S)$ be the set of all automorphisms on $S$. Consider a multiplicative function $\phi:T\to Aut(S)$, and define a **semidirect** product on $S\times T$ as below

$$(s_1,t_1)\circ(s_2,t_2)=\Big(s_1([\phi(t_1)](s_2)),t_1t_2\Big)$$ It was proven $S\times T$ with this product is semigroup. Also $S\times T$ with the following product is semigroup $$(s_1,t_1).(s_2,t_2)=(s_1s_2,t_1t_2)$$ It was proven that for any semicharacter $f$ on $(S\times T,.)$ there exists two unique semicharacters $g$ on $S$ and $h$ on $T$ such that for all $x\in S,y\in T$ $$f(x,y)=g(x)h(y)$$.

Now the question is this that what we can say about semicharacters on $(S\times T,\circ)$?