Strictly Piecewise Language is a subclass of Subregular Languages.
It is defined by a set of forbidden subsequences.
We can map the monoid of words to the monoid of sets of subsequences of length up to k.
$\begin{CD}
\Sigma^*\times \Sigma^* @>\circ>> \Sigma^* \\
@Vf \times fVV @VVfV \\
\mathbb{P}(\Sigma^{\leq k})\times \mathbb{P}(\Sigma^{\leq k}) @>\oplus>>\mathbb{P}(\Sigma^{\leq k})
\end{CD}$
$ \begin{CD}
(w_1,w_2) @>\circ>> w_1 \circ w_2 \\
@VP_{\leq k} \times P_{\leq k}VV @VVP_{\leq k}V \\
(P_{\leq k}(w_1),P_{\leq k}(w_2)) @>\oplus>>P_{\leq k}(w_1 \circ w_2)
\end{CD}\\$
$S_1 \oplus S_2=\cup(P_{\leq k}(S_1 \circ S_2))$
$P_{\leq k}(w_1 \circ w_2)=P_{\leq k}(w_1) \oplus P_{\leq k}(w_2) = \cup(P_{\leq k}(P_{\leq k}(w_1) \circ P_{\leq k}(w_2)))$