Three Notes on Controlled Hyper-Algebraic and Dhyper-Algebraic Extensions

(1) Regular control does not increase the generating power of 1-restricted [d]$K$-iteration grammars provided that $K\supseteq{\rm SYMBOL}$, and $K$ is closed under isomorphism and under union with ${\rm SYMBOL}$-languages. (2) Let $\Gamma$ be a prequasoid closed under the regular operations. If $K$ is a prequasoid [pseudoid], then $H(\Gamma)\subseteq H(\Gamma,K)$ [$\eta(\Gamma) \subseteq\eta(\Gamma,K)$. In particular we have $H(\Gamma) \subseteq (\Gamma){\rm ETOL}$ and $\eta(\Gamma) \subseteq (\Gamma){\rm EDTOL}$.(3) Underweak assumptions on $\Gamma$ and $K$, the decidability of the emptiness problem for $\Gamma$ and $K$ implies the decidability of the emptiness problem and the membership problem for the families $\eta(\Gamma,K)$ and $\eta(K)$.