  In this paper, we propose an extension of the standard strong and weak
lack-of-memory properties. We say that the survival function $\bar{F}$ of the
vector $(X,Y)$ satisfies pseudo lack-of-memory property in strong version if
\begin{equation} \label{strong}\bar F_{X,Y}(s_1+t_1,s_2+t_2)=\bar
F_{X,Y}(s_1,s_2)\otimes_h\bar F_{X,Y}(t_1,t_2), \ t_1,t_2,s_1,s_2 \geq 0
\end{equation} and in weak version if \begin{equation}\label{weak}\bar
F_{X,Y}(s_1+t,s_2+t)=\bar F_{X,Y}(s_1,s_2)\otimes_h\bar F_{X,Y}(t,t), \
s_1,s_2,t \geq 0\end{equation} with $a\otimes_hb=h\left (h^{-1}(a)\cdot
h^{-1}(b)\right )$, where $h$ is an increasing bijection of $[0,1]$, called
generator. After finding sufficient conditions under which the solutions of the
above functional equations are bivariate survival functions, we focus on
distributions satisfying the latter: we study specific properties in comparison
with standard lack-of-memory property and we give a characterization in terms
of the random variables $\min(X,Y)$ and $ X -Y$. Finally, we investigate the
induced dependence structure, determining their singularity in full generality
and studying the upper and lower dependence coefficients for some specific
choices of the marginal survival functions and of the generator $h$.
