rule under some assumptions. Here, the commutation rule is that a pair of operators $a$ and $b$ acting on a Hilbert space ${\cal H}$ satisfying $$\begin{aligned}
ab-ba=I. \nonumber\end{aligned}$$ The author has given assumptions to construct the regular biorthogonal pair in Ref. [@h-t2]. Indeed, the assumptions in Ref. [@h-t2] coincide with the definition of pseudo-bosons as originally given in Ref. [@bagarello10]. We shall give some assumptions to construct the semi-regular biorthogonal pair that connect with the definition of pseudo-bosons, and show that by using the results in Section 3 and Ref. [@h-t2], if $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then we may construct new pseudo-bosonic operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and if $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then we may construct a new pseudo-bosonic operators $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$. Furthermore, we investigate the relationship between pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ satisfying some assumptions and the operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$.

This article is organized as follows. In Section 2, we define new operators $T_{\phi,\bm{e}}$ and $T_{\bm{e},\phi}$ and study the property of these operators. Furthermore, we study the 